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A321600
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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 ).
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EXAMPLE
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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 + ...
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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