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%I #45 Jul 11 2013 15:06:17
%S 1,0,-1,0,0,1,4,0,0,-1,0,-5,0,0,1,0,0,6,0,0,-1,0,0,0,-7,0,0,1,0,-4,0,
%T 0,8,0,0,-1,9,0,9,0,0,-9,0,0,1,0,-10,0,-15,0,0,10,0,0,-1,0,0,11,0,22,
%U 0,0,-11,0,0,1,0,0,4,-12,0,-30,0,0,12,0,0,-1,0,-13,0,-13,13,0,39,0,0,-13,0,0,1,0,0,28,0,28,-14,0,-49,0,0,14,0,0,-1
%N Triangle generated by Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.
%H Paul D. Hanna, <a href="/A216273/b216273.txt">Rows n=1..45, flattened as a list of n, a(n) for n = 1..1035.</a>
%F G.f.: Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k*k*binomial(k+m,m)/n = 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1), which holds for all m >= 0.
%F Row sums equal A162552.
%F Sum_{k=1..n} T(n,k)*2^k = -(-1)^n*(sigma(2*n) - sigma(n)) for n>=1, where sigma is the sum of divisors of n, A000203.
%F Sum_{k=1..n} T(n,k)*2^k*k = -(-1)^n*n*A015128(n) for n>=1, where A015128(n) is the number of overpartitions of n, with g.f.: Product_{n>=1} (1+x^n)/(1-x^n).
%F Sum_{k=1..n} T(n,k)*2^k*k*(k+1) = -(-1)^n*4*n*A002318(n) for n>=1, where A002318 lists the coefficients in (1/theta_4(q)^2 -1)/4 in powers of q.
%F Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)/2! = -n*A004404(n) for n>=1, where A004404 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^3.
%F Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)*(k+3)/3! = -n*A004405(n) for n>=1, where A004405 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^4.
%F More generally:
%F Sum_{k=1..n} T(n,k)*y^k*k*binomial(k+m,m)/n = [x^n] 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1) for m>=0, n>=1.
%e G.f.: A(x,y) = y*x - y^2*x^2/2 + y^3*x^3/3 + (-y^4 + 4*y)*x^4/4 + (y^5 - 5*y^2)*x^5/5 + (-y^6 + 6*y^3)*x^6/6 + (y^7 - 7*y^4)*x^7/7 + (-y^8 + 8
%e *y^5 - 4*y^2)*x^8/8 + (y^9 - 9*y^6 + 9*y^3 + 9*y)*x^9/9 + (-y^10 + 10*y^7 - 15*y^4 - 10*y^2)*x^10/10 +...
%e where
%e exp(A(x,y)) = 1 + y*x + y*x^4 + y*x^9 + y*x^16 + y*x^25 +...
%e Triangle begins:
%e 1;
%e 0, -1;
%e 0, 0, 1;
%e 4, 0, 0, -1;
%e 0, -5, 0, 0, 1;
%e 0, 0, 6, 0, 0, -1;
%e 0, 0, 0, -7, 0, 0, 1;
%e 0, -4, 0, 0, 8, 0, 0, -1;
%e 9, 0, 9, 0, 0, -9, 0, 0, 1;
%e 0, -10, 0, -15, 0, 0, 10, 0, 0, -1;
%e 0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1;
%e 0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1;
%e 0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1;
%e 0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1;
%e 0, 0, 0, -45, 0, -50, 15, 0, 60, 0, 0, -15, 0, 0, 1;
%e 16, 0, 0, -4, 64, 0, 80, -16, 0, -72, 0, 0, 16, 0, 0, -1;
%e 0, -17, 17, 0, 17, -85, 0, -119, 17, 0, 85, 0, 0, -17, 0, 0, 1;
%e 0, -9, 18, -54, 0, -45, 108, 0, 168, -18, 0, -99, 0, 0, 18, 0, 0, -1;
%e 0, 0, 19, -19, 114, 0, 95, -133, 0, -228, 19, 0, 114, 0, 0, -19, 0, 0, 1;
%e 0, -20, 0, -30, 24, -200, 0, -175, 160, 0, 300, -20, 0, -130, 0, 0, 20, 0, 0, -1;
%e 0, 0, 42, -21, 42, -42, 315, 0, 294, -189, 0, -385, 21, 0, 147, 0, 0, -21, 0, 0, 1;
%e 0, 0, 22, -66, 88, -55, 88, -462, 0, -462, 220, 0, 484, -22, 0, -165, 0, 0, 22, 0, 0, -1;
%e 0, 0, 0, -69, 92, -230, 69, -184, 644, 0, 690, -253, 0, -598, 23, 0, 184, 0, 0, -23, 0, 0, 1;
%e 0, 0, 24, 0, 144, -124, 480, -84, 360, -864, 0, -990, 288, 0, 728, -24, 0, -204, 0, 0, 24, 0, 0, -1;
%e 25, -25, 0, -75, 25, -250, 175, -875, 100, -655, 1125, 0, 1375, -325, 0, -875, 25, 0, 225, 0, 0, -25, 0, 0, 1; ...
%o (PARI) {T(n,k)=n*polcoeff(polcoeff(log(1+sum(m=1,sqrtint(n)+1,y*x^(m^2))+x*O(x^n)),n,x),k,y)}
%o for(n=1,25,for(k=1,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Alternate g.f., true for all m >= 0: */
%o {T(n,k,m=0) = if(k<1|m<0,0, (n/k/binomial(k+m,m)) * polcoeff(polcoeff( 1 - 1/(1+sum(j=1,sqrtint(n+1),y*x^(j^2))+x*O(x^n))^(m+1), n,x),k,y))}
%o for(n=1, 25, for(k=1, n, print1(T(n, k,m=1), ", ")); print(""))
%Y Cf. A227311, A162552, A000203, A015128, A002318, A004404, A004405.
%K tabl,sign
%O 1,7
%A _Paul D. Hanna_, Mar 16 2013
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