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%I #32 May 02 2018 22:28:28
%S 1,0,1,1,0,0,6,6,0,0,0,4,56,56,4,0,0,0,1,117,722,722,117,1,0,0,0,0,
%T 126,2982,12012,12012,2982,126,0,0,0,0,0,84,6916,79548,246092,246092,
%U 79548,6916,84,0,0,0,0,0,36,10900,312880,2322000,6002824,6002824,2322000,312880,10900,36,0,0,0,0,0,9,12717,864009,13617765,74916306,170048394,170048394,74916306,13617765,864009,12717,9,0,0,0,0,0,1,11421,1825786,57282026,604000555,2669115383,5489377628,5489377628,2669115383,604000555,57282026,1825786,11421,1,0,0
%N G.f.: A(x,y) = (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2).
%C G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^n*y^k, where T(n,k) is the term of this triangle at position k in row n.
%H Paul D. Hanna, <a href="/A303920/b303920.txt">Table of n, a(n) for n = 0..2600, of rows 0..50, as a flattened triangle read by rows.</a>
%F GENERATING FUNCTIONS.
%F (1) A(x,y) = (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2).
%F (2) A(x,y) = (1-y) * Sum_{n>=0} y^n * q^n * Product_{k=1..n} (1 - q^(4*k-3)*y) / (1 - q^(4*k-1)*y), where q = 1 + x*(1-y)^2, due to a q-series identity.
%F (3) A(x,y) = (1-y)/(1 - q*y/(1 - q*(q^2-1)*y/(1 - q^5*y/(1 - q^3*(q^4-1)*y/(1 - q^9*y/(1- q^5*(q^6-1)*y/(1 - q^13*y/(1 - q^7*(q^8-1)*y/(1 - ...))))))))), where q = 1 + x*(1-y)^2, a continued fraction due to an identity of a partial elliptic theta function.
%F FORMULAS INVOLVING TERMS.
%F Sum_{k=0..2*n} T(n,k) = (2*n)!/n!, for n>=0 (row sums = A001813).
%F Sum_{k=0..2*n} T(n,k) * (-1)^k = 0, for n>=1 (symmetric rows).
%F Sum_{k=0..2*n} T(n,k) * 2^k = A265936(n), for n>=1.
%F Sum_{k=0..2*n} T(n,k) / 2^k = A173217(n) / 4^n, for n>=0.
%F Sum_{j=0..k^2} T(j,k) = A303922(k), for k>=0 (column sums).
%F T(n,n) = A303921(n), for n>=0 (diagonal).
%e G.f.: A(x,y) = (1-y) * ( 1 + y*(1 + x*(1-y)^2) + y^2*(1 + x*(1-y)^2)^4 + y^3*(1 + x*(1-y)^2)^9 + y^4*(1 + x*(1-y)^2)^16 + y^5*(1 + x*(1-y)^2)^25 + ... ).
%e Explicitly,
%e A(x,y) = 1 + x*(y + y^2) + x^2*(6*y^2 + 6*y^3) + x^3*(4*y^2 + 56*y^3 + 56*y^4 + 4*y^5) + x^4*(y^2 + 117*y^3 + 722*y^4 + 722*y^5 + 117*y^6 + y^7) + x^5*(126*y^3 + 2982*y^4 + 12012*y^5 + 12012*y^6 + 2982*y^7 + 126*y^8) + x^6*(84*y^3 + 6916*y^4 + 79548*y^5 + 246092*y^6 + 246092*y^7 + 79548*y^8 + 6916*y^9 + 84*y^10) + x^7*(36*y^3 + 10900*y^4 + 312880*y^5 + 2322000*y^6 + 6002824*y^7 + 6002824*y^8 + 2322000*y^9 + 312880*y^10 + 10900*y^11 + 36*y^12) + x^8*(9*y^3 + 12717*y^4 + 864009*y^5 + 13617765*y^6 + 74916306*y^7 + 170048394*y^8 + 170048394*y^9 + 74916306*y^10 + 13617765*y^11 + 864009*y^12 + 12717*y^13 + 9*y^14) + x^9*(y^3 + 11421*y^4 + 1825786*y^5 + 57282026*y^6 + 604000555*y^7 + 2669115383*y^8 + 5489377628*y^9 + 5489377628*y^10 + 2669115383*y^11 + 604000555*y^12 + 57282026*y^13 + 1825786*y^14 + 11421*y^15 + y^16) + ...
%e This triangle begins:
%e [1];
%e [0, 1, 1];
%e [0, 0, 6, 6, 0];
%e [0, 0, 4, 56, 56, 4, 0];
%e [0, 0, 1, 117, 722, 722, 117, 1, 0];
%e [0, 0, 0, 126, 2982, 12012, 12012, 2982, 126, 0, 0];
%e [0, 0, 0, 84, 6916, 79548, 246092, 246092, 79548, 6916, 84, 0, 0];
%e [0, 0, 0, 36, 10900, 312880, 2322000, 6002824, 6002824, 2322000, 312880, 10900, 36, 0, 0];
%e [0, 0, 0, 9, 12717, 864009, 13617765, 74916306, 170048394, 170048394, 74916306, 13617765, 864009, 12717, 9, 0, 0];
%e [0, 0, 0, 1, 11421, 1825786, 57282026, 604000555, 2669115383, 5489377628, 5489377628, 2669115383, 604000555, 57282026, 1825786, 11421, 1, 0, 0]; ...
%o (PARI) /* G.f. by Definition: */
%o {T(n,k) = my(A = (1-y) * sum(m=0,2*n, y^m * (1 + x*(1-y)^2 +x*O(x^(2*n)) )^(m^2))); polcoeff(polcoeff(A, n,x),k,y)}
%o for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))
%o (PARI) /* Continued fraction expression: */
%o {T(n,k) = my(CF=1, q = 1 + x*(1-y)^2 +x*O(x^(2*n))); for(k=0, n, CF = 1/(1 - q^(4*n-4*k+1)*y/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*y*CF)) ); polcoeff(polcoeff((1-y)*CF, n,x),k,y)}
%o for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))
%o (PARI) /* G.f. by q-series identity: */
%o {T(n,k) = my(A =1, q = 1 + x*(1-y)^2 +x*O(x^(2*n))); A = (1-y) * sum(m=0,2*n, y^m*q^m * prod(k=1,m, (1 - y*q^(4*k-3)) / (1 - y*q^(4*k-1) +x*O(x^(2*n))) )); polcoeff(polcoeff(A, n,x),k,y)}
%o for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", ")); print(""))
%Y Cf. A303921 (diagonal), A303922 (column sums), A001813 (row sums), A265936 (y=2), A173217.
%K nonn,tabf
%O 0,7
%A _Paul D. Hanna_, May 02 2018
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