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
Examples
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).
Links
Programs
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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(""))
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.
Extensions
More terms and b-file added by Paul D. Hanna, Jul 02 2016
Comments