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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

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

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, ...].

EXAMPLE

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.

...

PROG

(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(""))

CROSSREFS

Cf. A321600 (log).

Sequence in context: A279387 A309852 A029810 * A261122 A115660 A128581

Adjacent sequences:  A321598 A321599 A321600 * A321602 A321603 A321604

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 25 2018

STATUS

approved

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Last modified October 20 13:38 EDT 2021. Contains 348108 sequences. (Running on oeis4.)