A253383 - OEIS

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A253383

Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3k)^k.

1, 7, 2, 7, 38, 3, 7, 362, 111, 4, 7, 2522, 2271, 244, 5, 7, 14672, 34671, 8344, 455, 6, 7, 75908, 442911, 212464, 23135, 762, 7, 7, 361676, 5015199, 4498984, 869855, 53682, 1183, 8, 7, 1621388, 52044447, 83860840, 26997215, 2775282, 110047, 1736, 9, 7, 6935798, 505540767, 1423092160, 732435935, 117592782, 7458367, 205856, 2439, 10

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OFFSET

0,2

COMMENTS

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-3)^0 + T(n,1)*(x-3)^1 + T(n,2)*(x-6)^2 + ... + T(n,n)*(x-3n)^n, for n >= 0.

LINKS

Table of n, a(n) for n=0..54.

FORMULA

T(n,n-1) = n + 3*n^2 + 3*n^3, for n >= 1.

T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 12*n^2 - 6*n + 2)/2, for n >= 2.

T(n,n-3) = (n-2)*(9*n^6 - 54*n^5 + 81*n^4 + 9*n^3 - 12*n^2 - 45*n + 14)/2, for n >= 3.

EXAMPLE

The triangle T(n,k) starts:

n\k 0 1 2 3 4 5 6 7 8 ...

0: 1

1: 7 2

2: 7 38 3

3: 7 362 111 4

4: 7 2522 2271 244 5

5: 7 14672 34671 8344 455 6

6: 7 75908 442911 212464 23135 762 7

7: 7 361676 5015199 4498984 869855 53682 1183 8

8: 7 1621388 52044447 83860840 26997215 2775282 110047 1736 9

...

-----------------------------------------------------------------

n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 7*(x-0)^0 + 362*(x-3)^1 + 111*(x-6)^2 + 4*(x-9)^3.

PROG

(PARI) T(n, k)=(k+1)-sum(i=k+1, n, (-3*i)^(i-k)*binomial(i, k)*T(n, i))

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

CROSSREFS

Cf. A253382, A247237.

Sequence in context: A295421 A217172 A248791 * A010506 A197845 A082633

Adjacent sequences: A253380 A253381 A253382 * A253384 A253385 A253386

KEYWORD

nonn,tabl

AUTHOR

Derek Orr, Dec 30 2014

STATUS

approved