A136462 Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.

1, 1, 1, 1, 2, 1, 1, 3, 6, 4, 1, 4, 15, 56, 70, 1, 5, 28, 220, 1820, 4368, 1, 6, 45, 560, 10626, 201376, 906192, 1, 7, 66, 1140, 35960, 1712304, 74974368, 621216192, 1, 8, 91, 2024, 91390, 7624512, 927048304, 94525795200, 1429702652400, 1, 9, 120, 3276, 194580, 24040016, 5423611200, 1708566412608, 409663695276000, 11288510714272000, 1, 10, 153, 4960, 367290, 61124064, 21193254160, 13161885792000, 10895665708319184, 6208116950265950720, 312268282598377321216

Offset

0,5

Comments

Row n equals column 0 of matrix product A136467^(n+1) for n>=0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..495, for rows 0..30 of flattened table.

Formula

O.g.f. of row n: Sum_{k>=0} ((n+1)/2)^k * log(1 + 2^k*x)^k / k! = Sum_{k>=0} C((n+1)*2^(k-1), k) * x^k for n>=0.

Example

1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,...;
1,2,6,56,1820,201376,74974368,94525795200,409663695276000,...;
1,3,15,220,10626,1712304,927048304,1708566412608,...;
1,4,28,560,35960,7624512,5423611200,13161885792000,...;
1,5,45,1140,91390,24040016,21193254160,63815149590720,...;
1,6,66,2024,194580,61124064,64300886496,231207760388736,...;
1,7,91,3276,367290,134153712,163995687856,685581099291712,...;
1,8,120,4960,635376,264566400,368532802176,1756185841659392,...; ...
Triangle A136467 begins:
1;
1,1;
1,4,1;
4,32,16,1;
70,848,576,64,1;
4368,75648,62208,9216,256,1;
906192,22313216,21169152,3792896,143360,1024,1;
621216192,21827627008,23212261376,4793434112,223215616,2228224,4096,1;
such that row n of A136462 equals column 0 of A136467^(n+1).

Prog

(PARI) {T(n, k)=binomial((n+1)*2^(k-1), k)}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
(PARI) /* T(n, k) = Coefficient of x^k in series: */
{T(n, k)=polcoeff(sum(i=0, k, ((n+1)/2)^i*log(1+2^i*x +x*O(x^k))^i/i!), k)}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. rows: A136465, A014070, A136466, A101346; A136463 (diagonal); A136467.

Sequence in context: A376554 A056646 A056056 * A320574 A060517 A163181

Adjacent sequences: A136459 A136460 A136461 * A136463 A136464 A136465

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 31 2007

Extensions

More terms and b-file added by Paul D. Hanna, Jul 02 2016

Status

approved