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A321600 G.f. A(x,y) satisfies:  Sum_{n=-oo...+oo} (x^n + y)^n = exp( (1-y) * A(x,y) ) / (1-y), where A(x,y) = Sum_{n>=1} x^n/n * Sum{k=0..n-1} T(n,k)*y^k, written here as a flattened triangle of coefficients T(n,k) read by rows. 3
2, -4, 6, 8, -22, 26, -8, 64, -114, 78, 12, -148, 402, -478, 242, -16, 314, -1192, 2070, -1866, 726, 16, -614, 3110, -7334, 9578, -6886, 2186, -16, 1136, -7408, 22680, -39394, 41118, -24546, 6558, 26, -2008, 16694, -63526, 139730, -192622, 167426, -85294, 19682, -24, 3436, -35644, 165176, -444824, 768966, -880314, 655206, -290874, 59046, 24, -5718, 72910, -404402, 1303018, -2731958, 3902538, -3822454, 2486618, -977590, 177146, -32, 9292, -144086, 942646, -3571352, 8871544, -15280722, 18624942, -15945042, 9209454, -3247698, 531438, 28, -14766, 276070, -2108286, 9267104, -26804126, 54364754, -79614078, 84689282, -64399294, 33437042, -10687870, 1594322 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Related series identity: Sum_{n=-oo..+oo} (x^n + y)^n = Sum_{n=-oo..+oo} x^(n^2)/(1 - y*x^n)^(n+1).

See rectangle A321601 for other related identities.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1275 of the terms in this triangle as read by rows 1..50.

FORMULA

A(x,0) = log( 1 + 2*Sum_{n>=1} x^(n^2) ), the logarithm of the theta_3(x) series.

T(n,0) = (-1)^(n-1) * (sigma(2*n) - sigma(n)), for n >= 1.

Diagonal: Sum_{n>=1} T(n,n-1)*x^n/n = log( (1-x)*(1-x^2)/(1-3*x) ).

Row sums: Sum_{k=0..n-1} T(n,k) = n * A261608(n) for n >= 1, where A261608 is defined by g.f.: Sum_{n=-oo..+oo} (x^n + 1)^n (excluding coefficients of x^0).

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

EXAMPLE

GENERATING FUNCTION.

G.f.: A(x,y) = x*(2) + x^2*(-4 + 6*y)/2 + x^3*(8 - 22*y + 26*y^2)/3 + x^4*(-8 + 64*y - 114*y^2 + 78*y^3)/4 + x^5*(12 - 148*y + 402*y^2 - 478*y^3 + 242*y^4)/5 + x^6*(-16 + 314*y - 1192*y^2 + 2070*y^3 - 1866*y^4 + 726*y^5)/6 + x^7*(16 - 614*y + 3110*y^2 - 7334*y^3 + 9578*y^4 - 6886*y^5 + 2186*y^6)/7 + x^8*(-16 + 1136*y - 7408*y^2 + 22680*y^3 - 39394*y^4 + 41118*y^5 - 24546*y^6 + 6558*y^7)/8 + x^9*(26 - 2008*y + 16694*y^2 - 63526*y^3 + 139730*y^4 - 192622*y^5 + 167426*y^6 - 85294*y^7 + 19682*y^8)/9 + ...

such that

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

which begins

Sum_{n=-oo...+oo} (x^n + y)^n  =  1/(1-y) + 2*x + y*x^2 + 4*y^2*x^3 + (3*y^3 + 2)*x^4 + 6*y^4*x^5 + (5*y^5 + y)*x^6 + 8*y^6*x^7 + (7*y^7 + 9*y^2)*x^8 + (10*y^8 + 2)*x^9 + (9*y^9 + 6*y^3)*x^10 + 12*y^10*x^11 + (11*y^11 + 20*y^4 + y)*x^12 + 14*y^12*x^13 + (13*y^13 + 15*y^5)*x^14 + (16*y^14 + 16*y^2)*x^15 + (15*y^15 + 35*y^6 + 2)*x^16 + ...

Note the related series identity:

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

TRIANGLE OF COEFFICIENTS.

This triangle of coefficients T(n,k) of x^n*y^k/n in A(x,y) begins:

2;

-4, 6;

8, -22, 26;

-8, 64, -114, 78;

12, -148, 402, -478, 242;

-16, 314, -1192, 2070, -1866, 726;

16, -614, 3110, -7334, 9578, -6886, 2186;

-16, 1136, -7408, 22680, -39394, 41118, -24546, 6558;

26, -2008, 16694, -63526, 139730, -192622, 167426, -85294, 19682;

-24, 3436, -35644, 165176, -444824, 768966, -880314, 655206, -290874, 59046;

24, -5718, 72910, -404402, 1303018, -2731958, 3902538, -3822454, 2486618, -977590, 177146;

-32, 9292, -144086, 942646, -3571352, 8871544, -15280722, 18624942, -15945042, 9209454, -3247698, 531438;

28, -14766, 276070, -2108286, 9267104, -26804126, 54364754, -79614078, 84689282, -64399294, 33437042, -10687870, 1594322;

-32, 23040, -515050, 4550878, -22960886, 76301928, -179078456, 307580790, -392226346, 370301910, -253279146, 119416758, -34897962, 4782966; ...

in which the leftmost column equals (-1)^(n-1) * (sigma(2*n) - sigma(n)).

RELATED SERIES.

The g.f. A(x,0) of the leftmost column is given by

log( 1 + 2*Sum_{n>=1} x^(n^2) )  =  2*x - 4*x^2/2 + 8*x^3/3 - 8*x^4/4 + 12*x^5/5 - 16*x^6/6 + 16*x^7/7 - 16*x^8/8 + 26*x^9/9 - 24*x^10/10 + 24*x^11/11 - 32*x^12/12 + 28*x^13/13 - 32*x^14/14 + 48*x^15/15 - 32*x^16/16 + ... + A054785(n)*x^n/n + ...

The main diagonal may be generated by

log( (1-x)*(1-x^2)/(1-3*x) )  =  2*x + 6*x^2/2 + 26*x^3/3 + 78*x^4/4 + 242*x^5/5 + 726*x^6/6 + 2186*x^7/7 + 6558*x^8/8 + 19682*x^9/9 + 59046*x^10/10 + 177146*x^11/11 + 531438*x^12/12 + ... + A322116(n)*x^n/n + ...

where A322116(n) = T(n,n-1) for n >= 1.

The o.g.f. of the row sums is

Sum_{n=-oo..+oo} n^2 * x^n * (x^n + 1)^(n-1)  =  2*x + 2*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 36*x^6 + 56*x^7 + 128*x^8 + 108*x^9 + 150*x^10 + 132*x^11 + 384*x^12 + 182*x^13 + 392*x^14 + ... + n*A261608(n)*x^n + ...

At y = -1, we have the logarithmic series

A(x,-1) = 2*x - 10*x^2/2 + 56*x^3/3 - 264*x^4/4 + 1282*x^5/5 - 6184*x^6/6 + 29724*x^7/7 - 142856*x^8/8 + 687008*x^9/9 - 3303510*x^10/10 + 15884376*x^11/11 - 76378248*x^12/12 + ... + ( Sum_{k=0..n-1} T(n,k) * (-1)^k ) * x^n/n + ...

where (1/2) * exp( 2*A(x,-1) )  =  Sum_{n=-oo..+oo} (x^n - 1)^n - (-1)^n/2  =  1/2 + 2*x - x^2 + 4*x^3 - x^4 + 6*x^5 - 6*x^6 + 8*x^7 + 2*x^8 + 12*x^9 - 15*x^10 + 12*x^11 + 8*x^12 + ... + A261605(n)*x^n + ...

PROG

(PARI) {Q(m) = sum(n=-m-1, m+1, (x^n + y)^n +O(x^(m+2)))}

{T(n, k) = my(LOG=log((1-y)*Q(n) + y^(n+2))); n*polcoeff( polcoeff( LOG/(1-y), n, x), k, y)}

for(n=1, 16, for(k=0, n-1, print1( T(n, k), ", ")); print(""))

CROSSREFS

Cf. A321601, A261608, A261605, A054785, A322116.

Sequence in context: A045926 A045927 A165931 * A081939 A082615 A277258

Adjacent sequences:  A321597 A321598 A321599 * A321601 A321602 A321603

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Nov 21 2018

STATUS

approved

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Last modified October 21 11:51 EDT 2021. Contains 348150 sequences. (Running on oeis4.)