A321600 - OEIS

%I #53 Feb 24 2019 01:25:52

%S 2,-4,6,8,-22,26,-8,64,-114,78,12,-148,402,-478,242,-16,314,-1192,

%T 2070,-1866,726,16,-614,3110,-7334,9578,-6886,2186,-16,1136,-7408,

%U 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

%N 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.

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

%C See rectangle A321601 for other related identities.

%H Paul D. Hanna, <a href="/A321600/b321600.txt">Table of n, a(n) for n = 1..1275 of the terms in this triangle as read by rows 1..50.</a>

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

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

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

%F 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).

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

%e GENERATING FUNCTION.

%e 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 + ...

%e such that

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

%e which begins

%e 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 + ...

%e Note the related series identity:

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

%e TRIANGLE OF COEFFICIENTS.

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

%e 2;

%e -4, 6;

%e 8, -22, 26;

%e -8, 64, -114, 78;

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

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

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

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

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

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

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

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

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

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

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

%e RELATED SERIES.

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

%e 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 + ...

%e The main diagonal may be generated by

%e 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 + ...

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

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

%e 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 + ...

%e At y = -1, we have the logarithmic series

%e 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 + ...

%e 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 + ...

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

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

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

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

%K sign,tabl

%O 1,1

%A _Paul D. Hanna_, Nov 21 2018