A321601 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.

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Nov 25 2018

Keywords

• nonn
• tabl

Comments

Related triangle A321600 describes log( (1-y)*Sum_{n=-oo...+oo} (x^n + y)^n ) / (1-y).

Examples

			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) + ...
AS A RECTANGLE.
This sequence presents the coefficients of A(x,y) in the compact form
(*) A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k,
so that this table of coefficients T(n,k) of x^(n^2 + n*k) * y^k in A(x,y) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, ...;
2, 1, 9, 6, 20, 15, 35, 28, 54, 45, 77, 66, 104, ...;
2, 1, 16, 10, 50, 35, 112, 84, 210, 165, 352, 286, ...;
2, 1, 25, 15, 105, 70, 294, 210, 660, 495, 1287, 1001, ...;
2, 1, 36, 21, 196, 126, 672, 462, 1782, 1287, 4004, 3003, ...;
2, 1, 49, 28, 336, 210, 1386, 924, 4290, 3003, 11011, 8008, ...;
2, 1, 64, 36, 540, 330, 2640, 1716, 9438, 6435, 27456, ...;
2, 1, 81, 45, 825, 495, 4719, 3003, 19305, 12870, 63206, ...;
2, 1, 100, 55, 1210, 715, 8008, 5005, 37180, 24310, 136136, ...;
2, 1, 121, 66, 1716, 1001, 13013, 8008, 68068, 43758, 277134, ...;
...
where the g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.
Thus, for column k > 0,
if k odd: T(n,k) = binomial(n+k-1,k-1),
if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.
...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1325 of this rectangle as read by antidiagonals across rows 0..50

Crossrefs

Cf. A321600 (log).

Programs

PARI

/* Table of coefficients in Sum_{n=-oo...+oo} (x^n + y)^n */
ROWS=12
{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)}
{T(n,k) = polcoeff( sum(j=0, k, Q(n^2 + n*j,j)*z^j +O(z^(k+1))), k, z)}
for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))

PARI

/* Using binomial formula for terms T(n,k) */
ROWS=12
{T(n,k) = if(k==0, if(n==0,1,2),
if(k%2==1, binomial(n+k-1,k-1), binomial(n+k-1,k-1)*(2*n+k)/k))}
for(n=0, ROWS, for(k=0, ROWS, print1(T(n, k), ", ")); print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^(n^2 + n*k) * y^k satisfies

(1) A(x,0) = theta_3(x), a Jacobi theta function.

(2) A(x,y) = Sum_{n=-oo..+oo} (x^n + y)^n.

(3) A(x,y) = Sum_{n=-oo..+oo} x^(n^2)/(1 - x^n*y)^(n+1).

(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 ].

FORMULAS FOR TERMS.

The o.g.f. of column k is (1+x)/( (1 - (-1)^k*x) * (1-x)^k ) for k >= 0.

Thus, for column k > 0,

if k odd: T(n,k) = binomial(n+k-1,k-1),

if k even: T(n,k) = binomial(n+k-1,k-1)*(2*n+k)/k.

Antidiagonal sums are [1, 3, 4, 8, 16, ... , 2^n, ...].