A261605
G.f.: Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n)^n.
Original entry on oeis.org
1, 2, -1, 4, -1, 6, -6, 8, 2, 12, -15, 12, 8, 14, -28, 32, 22, 18, -55, 20, 34, 72, -66, 24, 44, 28, -91, 140, 62, 30, -205, 32, 209, 244, -153, 72, -98, 38, -190, 392, 443, 42, -518, 44, -1, 788, -276, 48, 506, 52, -451, 852, -196, 54, -1086, 728, 1636, 1180, -435, 60, -1691
Offset: 0
G.f.: A(x) = 1 + 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 + 14*x^13 - 28*x^14 + 32*x^15 + 22*x^16 +...
where A(x) = 1 + N(x) + P(x) such that
N(x) = (x-1) + (x^2-1)^2 + (x^3-1)^3 + (x^4-1)^4 + (x^5-1)^5 + (x^6-1)^6 +...
P(x) = x/(1-x) + x^4/(1-x^2)^2 + x^9/(1-x^3)^3 + x^16/(1-x^4)^4 + x^25/(1-x^5)^5 +...
explicitly,
N(x) = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 2*x^8 + 10*x^9 - 20*x^10 + 11*x^11 - x^12 + 13*x^13 - 35*x^14 + 25*x^15 + 13*x^16 +...
P(x) = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 5*x^10 + x^11 + 9*x^12 + x^13 + 7*x^14 + 7*x^15 + 9*x^16 +...+ A143862(n)*x^n +...
-
with(numtheory):
seq(add( (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1), d in divisors(n)), n = 1..60); # Peter Bala, Mar 03 2025
-
{a(n) = polcoeff(sum(m=-n-2,n+2,x^(m^2)/(1-x^m +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))
-
{a(n) = polcoeff(sum(m=-n-2,n+2,(x^m-1 +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))
-
{a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^(m^2)/(1+x^m +x*O(x^n))^(m+1)), n)}
for(n=0,60,print1(a(n),", "))
-
{a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^m*(1+x^m +x*O(x^n))^(m-1)), n)}
for(n=0,60,print1(a(n),", "))
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.
Original entry on oeis.org
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
Offset: 1
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 + ...
-
{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(""))
A292180
G.f.: Sum_{n=-oo..+oo} (1 + x^n)^n / (1 - x^n)^n, ignoring the constant term.
Original entry on oeis.org
4, 0, 16, 16, 24, 0, 32, 96, 116, 0, 48, 192, 56, 0, 608, 704, 72, 0, 80, 480, 1408, 0, 96, 3712, 2108, 0, 2720, 896, 120, 0, 128, 9600, 4672, 0, 17088, 12112, 152, 0, 7392, 20800, 168, 0, 176, 2112, 63032, 0, 192, 134400, 57828, 0, 15648, 2912, 216, 0, 130336, 69888, 21440, 0, 240, 317056, 248, 0, 556960, 428800, 282576, 0, 272, 4896, 36992, 0, 288, 1029600, 296, 0, 599024, 6080, 1859712, 0
Offset: 1
G.f.: A(x) = 4*x + 16*x^3 + 16*x^4 + 24*x^5 + 32*x^7 + 96*x^8 + 116*x^9 + 48*x^11 + 192*x^12 + 56*x^13 + 608*x^15 + 704*x^16 + 72*x^17 + 80*x^19 + 480*x^20 + 1408*x^21 + 96*x^23 + 3712*x^24 + 2108*x^25 + 2720*x^27 + 896*x^28 + 120*x^29 + 128*x^31 + 9600*x^32 + 4672*x^33 + 17088*x^35 + 12112*x^36 + 152*x^37 + 7392*x^39 + 20800*x^40 +...
where A(x) = Sum_{n=-oo..+oo} (1 + x^n)^n / (1 - x^n)^n, ignoring constant terms.
G.f. A(x) = P(x) + Q(x), where
P(x) = Sum_{n>=1} (1 + x^n)^n / (1 - x^n)^n - 1,
explicitly,
P(x) = 2*x + 6*x^2 + 8*x^3 + 18*x^4 + 12*x^5 + 44*x^6 + 16*x^7 + 66*x^8 + 58*x^9 + 92*x^10 + 24*x^11 + 276*x^12 + 28*x^13 + 156*x^14 + 304*x^15 + 386*x^16 + 36*x^17 + 674*x^18 + 40*x^19 + 1092*x^20 + 704*x^21 + 332*x^22 + 48*x^23 + 2852*x^24 + 1054*x^25 + 444*x^26 + 1360*x^27 + 3124*x^28 + 60*x^29 + 6648*x^30 + 64*x^31 + 4866*x^32 + 2336*x^33 + 716*x^34 + 8544*x^35 + 15494*x^36 + 76*x^37 + 876*x^38 + 3696*x^39 + 25796*x^40 +...
and
Q(x) = Sum_{n>=1} (-1)^n * (1 - x^n)^n / (1 + x^n)^n - (-1)^n,
explicitly,
Q(x) = 2*x - 6*x^2 + 8*x^3 - 2*x^4 + 12*x^5 - 44*x^6 + 16*x^7 + 30*x^8 + 58*x^9 - 92*x^10 + 24*x^11 - 84*x^12 + 28*x^13 - 156*x^14 + 304*x^15 + 318*x^16 + 36*x^17 - 674*x^18 + 40*x^19 - 612*x^20 + 704*x^21 - 332*x^22 + 48*x^23 + 860*x^24 + 1054*x^25 - 444*x^26 + 1360*x^27 - 2228*x^28 + 60*x^29 - 6648*x^30 + 64*x^31 + 4734*x^32 + 2336*x^33 - 716*x^34 + 8544*x^35 - 3382*x^36 + 76*x^37 - 876*x^38 + 3696*x^39 - 4996*x^40 +...
Terms at square positions divided by 4 begin:
a(n^2)/4 = [1, 4, 29, 176, 527, 3028, 14457, 107200, 446745, 2392604, 13286165, 140564336, 415382567, 2333455268, 17078911507, 78663453440, 419472490547, 2377516612900, 13482186743565, 78663154105296, 437169506932981, 2481447593907572, 14146164790774889, 161511806183206336, 460995825168188653, 2634869356953946428, 15071070681878977525, 86632929673574593072, 494051395886263605335, 2955861929786748934348, 16234283204352299108321, ...].
-
{a(n) = my(A,Ox=x*O(x^n)); A = sum(n=-n-1,n+1, if(n==0,0, (1 + x^n +Ox)^n/(1-x^n +Ox)^n - 1/2 +Ox )); polcoeff(A,n)}
for(n=1,80,print1(a(n),", "))
-
{a(n) = my(A,Ox=x*O(x^n)); A = sum(m=1,n+1, ((1+x^m +Ox)^(2*m) + (-1)^m*(1 - x^m +Ox)^(2*m))/(1 - x^(2*m) +Ox)^m - 1 ); polcoeff(A,n)}
for(n=1,80,print1(a(n),", "))
Showing 1-3 of 3 results.
Comments