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%I #51 Nov 27 2018 18:07:03
%S 1,2,1,2,1,1,2,1,4,1,2,1,9,3,1,2,1,16,6,6,1,2,1,25,10,20,5,1,2,1,36,
%T 15,50,15,8,1,2,1,49,21,105,35,35,7,1,2,1,64,28,196,70,112,28,10,1,2,
%U 1,81,36,336,126,294,84,54,9,1,2,1,100,45,540,210,672,210,210,45,12,1,2,1,121,55,825,330,1386,462,660,165,77,11,1,2,1,144,66,1210,495,2640,924,1782,495,352,66,14,1,2,1,169,78,1716,715,4719,1716,4290,1287,1287,286,104,13,1,2,1,196,91,2366,1001,8008,3003,9438,3003,4004,1001,546,91,16,1,2,1,225,105,3185,1365,13013,5005,19305,6435,11011,3003,2275,455,135,15,1
%N G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k, written here as a rectangle of coefficients T(n,k) read by antidiagonals.
%C Related triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n ) / (1-y).
%H Paul D. Hanna, <a href="/A321601/b321601.txt">Table of n, a(n) for n = 0..1325 of this rectangle as read by antidiagonals across rows 0..50</a>
%F G.f. A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k satisfies
%F (1) A(x,0) = theta_3(x), a Jacobi theta function.
%F (2) A(x,y) = Sum_{n=-oo..+oo} (x^n + y)^n.
%F (3) A(x,y) = Sum_{n=-oo..+oo} x^(n^2)/(1 - x^n*y)^(n+1).
%F (4) A(x,y) = theta_3(x) + Sum_{n>=0} Sum_{k>=0} x^(n*(n+2*k+1)) * y^(2*k+1) * [ binomial(n+2*k,2*k) + binomial(n+2*k+1,2*k+1)*(n+k+1)/(k+1) * x^n*y ].
%F FORMULAS FOR TERMS.
%F The o.g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.
%F Thus, for column k > 0,
%F if k odd: T(n,k) = binomial(n+k-1,k-1),
%F if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.
%F Antidiagonal sums are [1, 3, 4, 8, 16, ... , 2^n, ...].
%e G.f.: A(x,y) = Sum_{n=-oo...+oo} (x^n + y)^n = 1/(1 - y) + x*(2) + x^2*(y) + x^3*(4*y^2) + x^4*(2 + 3*y^3) + x^5*(6*y^4) + x^6*(y + 5*y^5) + x^7*(8*y^6) + x^8*(9*y^2 + 7*y^7) + x^9*(2 + 10*y^8) + x^10*(6*y^3 + 9*y^9) + x^11*(12*y^10) + x^12*(y + 20*y^4 + 11*y^11) + x^13*(14*y^12) + x^14*(15*y^5 + 13*y^13) + x^15*(16*y^2 + 16*y^14) + x^16*(2 + 35*y^6 + 15*y^15) + x^17*(18*y^16) + x^18*(10*y^3 + 28*y^7 + 17*y^17) + x^19*(20*y^18) + x^20*(y + 54*y^8 + 19*y^19) + x^21*(50*y^4 + 22*y^20) + x^22*(45*y^9 + 21*y^21) + x^23*(24*y^22) + x^24*(25*y^2 + 35*y^5 + 77*y^10 + 23*y^23) + x^25*(2 + 26*y^24) + x^26*(66*y^11 + 25*y^25) + x^27*(112*y^6 + 28*y^26) + x^28*(15*y^3 + 104*y^12 + 27*y^27) + x^29*(30*y^28) + x^30*(y + 84*y^7 + 91*y^13 + 29*y^29) + x^31*(32*y^30) + x^32*(105*y^4 + 135*y^14 + 31*y^31) + x^33*(210*y^8 + 34*y^32) + x^34*(120*y^15 + 33*y^33) + x^35*(36*y^2 + 36*y^34) + x^36*(2 + 70*y^5 + 165*y^9 + 170*y^16 + 35*y^35) + ...
%e AS A RECTANGLE.
%e This sequence presents the coefficients of A(x,y) in the compact form
%e (*) A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k,
%e so that this table of coefficients T(n,k) of x^(n^2 + n*k) * y^k in A(x,y) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, ...;
%e 2, 1, 9, 6, 20, 15, 35, 28, 54, 45, 77, 66, 104, ...;
%e 2, 1, 16, 10, 50, 35, 112, 84, 210, 165, 352, 286, ...;
%e 2, 1, 25, 15, 105, 70, 294, 210, 660, 495, 1287, 1001, ...;
%e 2, 1, 36, 21, 196, 126, 672, 462, 1782, 1287, 4004, 3003, ...;
%e 2, 1, 49, 28, 336, 210, 1386, 924, 4290, 3003, 11011, 8008, ...;
%e 2, 1, 64, 36, 540, 330, 2640, 1716, 9438, 6435, 27456, ...;
%e 2, 1, 81, 45, 825, 495, 4719, 3003, 19305, 12870, 63206, ...;
%e 2, 1, 100, 55, 1210, 715, 8008, 5005, 37180, 24310, 136136, ...;
%e 2, 1, 121, 66, 1716, 1001, 13013, 8008, 68068, 43758, 277134, ...;
%e ...
%e where the g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.
%e Thus, for column k > 0,
%e if k odd: T(n,k) = binomial(n+k-1,k-1),
%e if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.
%e ...
%o (PARI) /* Table of coefficients in Sum_{n=-oo...+oo} (x^n + y)^n */
%o ROWS=12
%o {Q(n, k) = polcoeff(polcoeff( sum(m=-n-k, n+k, (x^m + y +O(x^(n+1)))^m ), n, x) +O(y^(k+1)), k, y)}
%o {T(n,k) = polcoeff( sum(j=0, k, Q(n^2 + n*j,j)*z^j +O(z^(k+1))), k, z)}
%o for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))
%o (PARI) /* Using binomial formula for terms T(n,k) */
%o ROWS=12
%o {T(n,k) = if(k==0, if(n==0,1,2),
%o if(k%2==1, binomial(n+k-1,k-1), binomial(n+k-1,k-1)*(2*n+k)/k))}
%o for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))
%Y Cf. A321600 (log).
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Nov 25 2018
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