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%I #25 Mar 03 2018 12:36:58
%S 1,0,1,0,3,1,0,5,10,1,0,7,57,21,1,0,9,252,246,36,1,0,11,969,2158,710,
%T 55,1,0,13,3414,15927,10260,1635,78,1,0,15,11329,104883,122125,35085,
%U 3255,105,1,0,17,35992,637252,1273192,611130,96992,5852,136,1,0,19,110625,3647268,12057412,9199386,2321004,230972,9756,171,1,0,21,331298,19935477,106181320,124315310,47518716,7261394,492408,15345,210,1,0,23,971609,105054633,883422885,1546241270,865414802,193797618,19669302,963795,23045,253,1
%N Triangle defined by T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), for n>=0, k = 0..n, as read by rows.
%C Is there a closed-form expression for the terms T(n,k) of this triangle?
%C Row sums form A300279, with g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
%H Paul D. Hanna, <a href="/A300280/b300280.txt">Table of n, a(n) for n = 0..1080, as a flattened triangle read by Rows 0..45</a>
%F T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1).
%F G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n * y^k is given by:
%F (1) A(x,y) = Sum_{n>=0} (1 + x*y * (1+x)^n)^n / 2^(n+1).
%F (2) A(x,y) = Sum_{n>=0} x^n * y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).
%e This triangle begins:
%e 1;
%e 0, 1;
%e 0, 3, 1;
%e 0, 5, 10, 1;
%e 0, 7, 57, 21, 1;
%e 0, 9, 252, 246, 36, 1;
%e 0, 11, 969, 2158, 710, 55, 1;
%e 0, 13, 3414, 15927, 10260, 1635, 78, 1;
%e 0, 15, 11329, 104883, 122125, 35085, 3255, 105, 1;
%e 0, 17, 35992, 637252, 1273192, 611130, 96992, 5852, 136, 1;
%e 0, 19, 110625, 3647268, 12057412, 9199386, 2321004, 230972, 9756, 171, 1;
%e 0, 21, 331298, 19935477, 106181320, 124315310, 47518716, 7261394, 492408, 15345, 210, 1;
%e 0, 23, 971609, 105054633, 883422885, 1546241270, 865414802, 193797618, 19669302, 963795, 23045, 253, 1; ...
%e GENERATING FUNCTIONS.
%e G.f.: A(x,y) = Sum_{n>=0} x^n*y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).
%e Expanding,
%e G.f.: A(x,y) = 1 + x*y*(1+x)/(2 - (1+x))^2 + x^2*y^2*(1+x)^4/(2 - (1+x)^2)^3 + x^3*y^3*(1+x)^9/(2 - (1+x)^3)^4 + x^4*y^4*(1+x)^16/(2 - (1+x)^4)^5 + x^5*y^5*(1+x)^25/(2 - (1+x)^5)^6 + x^6*y^6*(1+x)^36/(2 - (1+x)^6)^7 + ...
%e Also, due to a series identity:
%e A(x,y) = 1/2 + (1 + x*y*(1+x))/2^2 + (1 + x*y*(1+x)^2)^2/2^3 + (1 + x*y*(1+x)^3)^3/2^4 + (1 + x*y*(1+x)^4)^4/2^5 + (1 + x*y*(1+x)^5)^5/2^6 + (1 + x*y*(1+x)^6)^6/2^7 + ... + (1 + x*y * (1+x)^n)^n / 2^(n+1) + ...
%e Explicitly,
%e G.f.: A(x,y) = 1 + y*x + (y^2 + 3*y)*x^2 + (y^3 + 10*y^2 + 5*y)*x^3 + (y^4 + 21*y^3 + 57*y^2 + 7*y)*x^4 + (y^5 + 36*y^4 + 246*y^3 + 252*y^2 + 9*y)*x^5 + (y^6 + 55*y^5 + 710*y^4 + 2158*y^3 + 969*y^2 + 11*y)*x^6 + (y^7 + 78*y^6 + 1635*y^5 + 10260*y^4 + 15927*y^3 + 3414*y^2 + 13*y)*x^7 + (y^8 + 105*y^7 + 3255*y^6 + 35085*y^5 + 122125*y^4 + 104883*y^3 + 11329*y^2 + 15*y)*x^8 + ...
%e The row sums begin
%e A300279 = [1, 1, 4, 16, 86, 544, 3904, 31328, 276798, 2660564, ...],
%e and has g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
%e RELATED TRIANGLE.
%e The coefficients in 1/A(x,y) forms the triangle:
%e 1;
%e 0, -1;
%e 0, -3, 0;
%e 0, -5, -4, 0;
%e 0, -7, -38, -4, 0;
%e 0, -9, -208, -104, -4, 0;
%e 0, -11, -884, -1336, -202, -4, 0;
%e 0, -13, -3268, -12112, -4768, -332, -4, 0;
%e 0, -15, -11098, -89540, -75532, -12520, -494, -4, 0; ...
%o (PARI) /* Must set N to a large value for accuracy: */ N=10000;
%o {T(n,k) = round( sum(j=0,N, binomial(j+k,k) * binomial((j+k)*k,n-k) / 2^(j+k+1)*1. ) )}
%o for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
%o (PARI) /* Faster, without precision errors: */
%o {T(n,k) = my(A = sum(m=0, n, x^m * y^m * (1+x + x*O(x^n))^(m^2) / (2 - (1+x + x*O(x^n))^m )^(m+1) )); polcoeff(polcoeff(A, n,x), k,y)}
%o for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
%Y Cf. A300279 (row sums).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Mar 01 2018
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