A181544 - OEIS

A181544

Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3*x^k]*(1-x)^(3n+1), read by rows of k=0..2n terms.

1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 54, 405, 760, 405, 54, 1, 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1, 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1, 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1, 1, 490, 35623, 818300, 7917371, 37215794, 91789005, 123519792, 91789005, 37215794, 7917371, 818300, 35623, 490, 1, 1, 704, 73200, 2430400, 34657700, 246781248, 955910032, 2116980800, 2751843600, 2116980800, 955910032, 246781248, 34657700, 2430400, 73200, 704, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1088, as a flattened triangle of rows 0..32

Ilia Gaiur, Vladimir Rubtsov, and Duco van Straten, Product formulas for the Higher Bessel functions, arXiv:2405.03015 [math.AG], 2024. See p. 18.

FORMULA

Row sums equal A006480(n) = (3n)!/(n!)^3, which is de Bruijn's s(3,n).

From Yahia Kahloune, Jan 30 2014: (Start)

Using these coefficients we can obtain formulas for the sums

Sum_{i=1..n} C(e-1+i,e)^3. Let us define b(k,e,3) = sum_{i=0..k-e} (-1)^i*C(3*e+1,i)*C(k-i,e)^3, where k=e+i.

For example:

b(e,e,3) = 1;

b(e+1,e,3) = (e+1)^3-(3*e+1) = e^2*(e+3);

b(e+2,e,3) = C(e+2,2)^3 - (3*e+1)*(e+1)^3 + C(3*e+1,2);

b(e+3,e,3) = C(e+3,e)^3 - (3*e+1)*C(e+2,e)^3 + C(3*e+1,2)*C(e+1,e)^3 - C(3*e+1,3);

b(e+4,e,3) = C(e+4,e)^3 - (3*e+1)*C(e+3,e)^3 + C(3*e+1,2)*C(e+2,e) - C(3*e+1,3)*C(e+1,e)^3 + C(3*e+1,4).

Then we have the formula: Sum_{i=1..n} C(e-1+i,e)^3 = Sum_{i=0..2*e} b(e+i,e,3)*C(n+e+i,3*e+1).

Example: Sum_{i=1..7} C(2+i,3)^3 = C(10,10) + 54*C(11,10) + 405*C(12,10) + 760*C(13,10) + 405*C(14,10) + 54*C(15,10) + C(16,10) = 820260.

(End)

EXAMPLE

Triangle begins:

1, 4, 1;

1, 20, 48, 20, 1;

1, 54, 405, 760, 405, 54, 1;

1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1;

1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1;

1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1; ...

Row g.f.s begin:

(1) = (1-x)*(1 + x + x^2 + x^3 + x^4 +...);

(1 + 4*x + x^2) = (1-x)^4*(1 + 2^3*x + 3^3*x^2 + 4^3*x^3 +...);

(1 + 20*x + 48*x^2 + 20*x^3 + x^4) = (1-x)^7*(1 + 3^3*x + 6^3*x^2 +...);

(1 + 54*x + 405*x^2 + 760*x^3 + 405*x^4 + 54*x^5 + x^6) = (1-x)^10*(1 + 4^3*x + 10^3*x^2 + 20^3*x^3 + 35^3*x^4 +...); ...

MATHEMATICA

t[n_, k_] := SeriesCoefficient[Sum[Binomial[n+j, j]^3*x^j, {j, 0, n+k}]*(1-x)^(3*n+1), {x, 0, k}]; Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Feb 04 2014, after PARI *)

PROG

(PARI) {T(n, k)=polcoeff(sum(j=0, n+k, binomial(n+j, j)^3*x^j)*(1-x)^(3*n+1), k)}

for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A181543, A181545, A006480, A002897, A000172.

Cf. A183204 (central terms), A183205.

Sequence in context: A340476 A176422 A156586 * A154283 A185946 A015113

Adjacent sequences: A181541 A181542 A181543 * A181545 A181546 A181547

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Oct 30 2010

STATUS

approved