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Expansion of e.g.f. 1/(1 - x^2 * exp(x))^3.
+0
2
1, 0, 6, 18, 180, 1500, 15930, 191646, 2580648, 38683224, 636068430, 11392350090, 220658360076, 4594593295188, 102333126352002, 2427278515815510, 61079333377870800, 1625065147997303856, 45576552142354413078, 1343802083242003570818, 41552482139458105525620
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(k+2,2)/(n-2*k)!.
a(n) ~ n! * n^2 / ((1 + LambertW(1/2))^3 * 2^(n+4) * LambertW(1/2)^n). - Vaclav Kotesovec, Oct 31 2024
PROG
(PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(k+2, 2)/(n-2*k)!);
E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.
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2
1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217
FORMULA
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));
E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x)^2)^2.
+0
2
1, 2, 26, 618, 22256, 1081770, 66401532, 4931389358, 430108545680, 43104305664594, 4881518010253460, 616559703960596022, 85935621525038617752, 13102417265843584412474, 2169337115977056447577820, 387609934848899388554651550, 74340899731294447790784890912
FORMULA
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377526. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+1,k)/( (2*k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+1, k)/((2*k+1)*(n-k)!));
E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.
+0
3
1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
FORMULA
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));
E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.
+0
3
1, 3, 36, 735, 21972, 871995, 43308378, 2588123811, 180990517032, 14507325973395, 1311719669172750, 132102208441613883, 14666354372331521676, 1779817542971018697003, 234399632982398657764578, 33297612755940733707395955, 5075234637265322738651060688, 826215756199826873368252279971
FORMULA
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364987. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+2,k)/( (k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+2, k)/((k+1)*(n-k)!));
E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^2.
+0
4
1, 2, 18, 270, 5936, 173330, 6335772, 278724362, 14350790064, 847007698338, 56397332340020, 4182866692785242, 342022887565717800, 30570009715185100082, 2965368922693150575084, 310276298423966343555690, 34834957115496822249510752, 4177193847524372747798263106
FORMULA
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364983. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+1, k)/((k+1)*(n-k)!));
Indices where new terms arise among first differences of Colombian or self numbers ( A377472).
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2
1, 5, 103, 984, 9785, 97786, 977787, 9777788
COMMENTS
See A377473 for the distinct values of the first differences in the order they appear for the first time.
FORMULA
a(n+1) = a(n) + 88*10^(n-1) + 1 for n = 3, 4, ..., 7 at least.
EXAMPLE
The first value, A377472(1) = 2, appears obviously at index a(1) = 1. The next three values are the same, but at index a(2) = 5 we have a new, distinct value A377472(5) = 11 = A377423(2). The next distinct value is A377472(103) = 15 = A377423(3), so a(3) = 103. Then the next new value is A377472(984) = 28 = A377423(4), so a(4) = 984. The next new value is A377472(9785) = 41 = A377423(5), so a(5) = 9785. Only at n = 977787 = a(7), we have a new value, A377472(n) = 67 = A377423(7). At n = 9777788 = a(8), we have the next new value, A377472(n) = 80 = A377423(8).
PROG
(PARI) A377473_upto(N=9, show=1)={my(o, c, d, L=List()); for(n=1+o=1, oo, is_ A003052(n)||next; c++; if(!setsearch(L, d=n-o), show && printf("%d, ", [c, d]); listput(L, c); #L<N||break); o=n); L}
Distinct first differences of Colombian or self numbers ( A377472), listed in the order they appear.
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2
2, 11, 15, 28, 41, 54, 67, 80
COMMENTS
See A377474 for the indices where these first differences appear for the first time.
EXAMPLE
A377472(n) = 2 = a(1) for all n <= 4. Then, A377472(n) = 11 = a(2) up to n = 13.
Then again, A377472(14..23) = (2, 11, ..., 11) and similarly up to n = 94. But A377472(103) = 15 = a(3). Then the previous pattern repeats, with A377472(n) = 2 for n = 112, 122, ..., 192, followed by A377472(n) = 15 at n = 201, 299, 397, ..., 887. Then A377472(984) = 28 = a(4), and it goes on with A377472(n) = 2 at n = 992, 1002, ..., 1072, and so on, with A377472(n) = 28 at n = 1962, 2940, 3918, ..., 8808. Then A377472(9785) = 41 = a(5), and the whole previous pattern repeats, with A377472(9881) = 15, then A377472(10762) = 28 etc. At n = 97786, we find A377472(n) = 54 = a(6), and again the whole previous pattern repeats again 8 more times, each time separated by a 54, until we have, at n = 977787, A377472(n) = 67 = a(7). And so on.
PROG
(PARI) A377473_upto(N=9, show=1)={my(o, c, d, L=List()); for(n=1+o=1, oo, is_ A003052(n)||next; c++; if(!setsearch(L, d=n-o), show && printf("%d, ", [c, d]); listput(L, d); #L<N||break); o=n); L}
CROSSREFS
Cf. A003052 (Colombian numbers), A377472 (1st differences of Colombian numbers).
Number of odd terms in the Collatz trajectory of k = 4n-1 which are a new record high among its odd terms.
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0
1, 2, 1, 3, 1, 2, 14, 13, 1, 3, 1, 12, 1, 12, 2, 11, 1, 11, 1, 3, 11, 2, 11, 10, 1, 10, 10, 10, 1, 2, 2, 6, 1, 2, 1, 9, 10, 2, 9, 8, 1, 9, 9, 8, 1, 8, 2, 5, 8, 9, 1, 8, 1, 8, 3, 7, 1, 8, 9, 7, 8, 2, 8, 7, 8, 7, 1, 3, 7, 2, 7, 4, 7, 3, 8, 3, 1, 7, 2, 6, 7, 8, 1, 6, 4, 6, 7, 6, 1, 6, 1, 4, 1, 2, 3, 6, 7, 6, 6, 6
COMMENTS
For these k = 4n-1, the first odd number is A139391(k) > k so it is the first record high. The trajectory of k starts with A001511(n) successive initial records, so that a(n) >= A001511(n) (and reaches A351974(n) at that point).
EXAMPLE
For n = 7, which corresponds to the Collatz trajectory started from 27, the trajectory reaches larger maximum odd numbers at the following points: 41, 47, 71, 107, 233, 263, 395, 593, 719, 1079, 1619, 2429, 3077. Since there are 14 instances where a new maximum odd number is reached, we have a(7)=14.
PROG
(Python)
def a(num:int) -> int:
count = 0
num = num * 4 - 1
maxnum = num
while num > 1:
if num%2 == 1:
num = num*3 + 1
while num%2 == 0:
num //= 2
if num > maxnum:
count += 1
maxnum = num
return count
Total number of ways a triangle of order n can be completely surrounded by a specific pentiamond.
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1
COMMENTS
The coronal expansion number is defined to be the number of expansions the central host frame can undergo and still have a corona formed by the coronal tile. The coronal expansion number is 4 for this sequence. See the links section for an example of a pseudo-triangle as the central host frame that has a coronal expansion number of 7. Craig Knecht, Oct 31 2024
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