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%I #11 Jul 02 2016 08:10:55
%S 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,
%T 10626,201376,906192,1,7,66,1140,35960,1712304,74974368,621216192,1,8,
%U 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
%N Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.
%C Row n equals column 0 of matrix product A136467^(n+1) for n>=0.
%H Paul D. Hanna, <a href="/A136462/b136462.txt">Table of n, a(n) for n = 0..495, for rows 0..30 of flattened table.</a>
%F 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.
%e 1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,...;
%e 1,2,6,56,1820,201376,74974368,94525795200,409663695276000,...;
%e 1,3,15,220,10626,1712304,927048304,1708566412608,...;
%e 1,4,28,560,35960,7624512,5423611200,13161885792000,...;
%e 1,5,45,1140,91390,24040016,21193254160,63815149590720,...;
%e 1,6,66,2024,194580,61124064,64300886496,231207760388736,...;
%e 1,7,91,3276,367290,134153712,163995687856,685581099291712,...;
%e 1,8,120,4960,635376,264566400,368532802176,1756185841659392,...; ...
%e Triangle A136467 begins:
%e 1;
%e 1,1;
%e 1,4,1;
%e 4,32,16,1;
%e 70,848,576,64,1;
%e 4368,75648,62208,9216,256,1;
%e 906192,22313216,21169152,3792896,143360,1024,1;
%e 621216192,21827627008,23212261376,4793434112,223215616,2228224,4096,1;
%e such that row n of A136462 equals column 0 of A136467^(n+1).
%o (PARI) {T(n,k)=binomial((n+1)*2^(k-1),k)}
%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%o (PARI) /* T(n,k) = Coefficient of x^k in series: */
%o {T(n,k)=polcoeff(sum(i=0,k,((n+1)/2)^i*log(1+2^i*x +x*O(x^k))^i/i!),k)}
%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%Y Cf. rows: A136465, A014070, A136466, A101346; A136463 (diagonal); A136467.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Dec 31 2007
%E More terms and b-file added by _Paul D. Hanna_, Jul 02 2016
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