A261605 -id:A261605 - cp's OEIS frontend

A260147 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n).

1, 2, 1, 5, 1, 6, 8, 8, 1, 25, 12, 12, 29, 14, 36, 77, 1, 18, 151, 20, 71, 135, 166, 24, 121, 236, 287, 307, 30, 30, 1141, 32, 1, 727, 681, 1247, 314, 38, 970, 1652, 1821, 42, 2633, 44, 331, 6590, 1772, 48, 497, 3053, 7146, 6801, 1717, 54, 4051, 7427, 8009, 12389, 3655, 60, 17842, 62, 4496, 42841, 1, 15731, 6470, 68, 19449, 34754, 65781

Offset: 0

Paul D. Hanna, Jul 17 2015

Compare to the curious identities:
(1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
(2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n  =  0.
Name changed for clarity by Paul D. Hanna, Dec 10 2024; prior name was "G.f.: (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function."

			G.f.: A(x) = 1 + 2*x^2 + x^4 + 5*x^6 + x^8 + 6*x^10 + 8*x^12 + 8*x^14 + x^16 + 25*x^18 + 12*x^20 + ...
where 2*A(x) = 1 + P(x) + N(x) with
P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 + ...
N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 + ...
Explicitly,
P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n + ...
N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n + ...
From _Paul D. Hanna_, Dec 10 2024: (Start)
SPECIFIC VALUES.
A(z) = 0 at z = -0.404783857785183643579648014798209689698619095608142590080356...
  where 0 = Sum_{n=-oo..+oo} z^n * (1 + z^n)^(2*n).
A(t) = 8 at t = 0.66184860446935243758952792459096102121713616089603...
A(t) = 7 at t = 0.64280265347584821638335226655422639958638446962646...
A(t) = 6 at t = 0.61846293982236470622283664293769398297407552626520...
A(t) = 5 at t = 0.58591538561726828976301449562779896617926938759041...
A(t) = 4 at t = 0.53948212974289878102531393938569583066950526874204...
A(t) = 3 at t = 0.46633361832235508894561538442655261465230172977527...
A(t) = 2 at t = 0.33014122063168294490173944063355594394361494532642...
  where 2 = Sum_{n=-oo..+oo} t^n * (1 + t^n)^(2*n).
A(t) = -1 at t = -0.57221202613754835881500708971837082259712665852148...
A(t) = -2 at t = -0.66124771863833308133360587362156745037996654826889...
A(t) = -3 at t = -0.72841228559829175547612598129696947453305714538354...
A(t) = -4 at t = -0.90975449896027994776675798799643226140294233213401...
A(4/5) = 39.597156112579883800797829785472315940190856875500...
A(3/4) = 18.522637966827153559321082877260756270457362912092...
A(2/3) = 8.2917909754417331599016245586686519315443444070756...
A(3/5) = 5.3942577326786364433206097043093210828422082884565...
A(1/2) = 3.3971121875472777749836900920631175982646917998641...
  where A(1/2) = Sum_{n=-oo..+oo} (2^n + 1)^(2*n) / 2^(2*n^2+n).
A(2/5) = 2.4226617866265771206729430879848898772232404418272...
A(1/3) = 2.0164022766484546805373278337731916678136050742206...
  where A(1/3) = Sum_{n=-oo..+oo} (3^n + 1)^(2*n) / 3^(2*n^2+n).
A(1/4) = 1.6529591092151291503041860933179009814428123139546...
  where A(1/4) = Sum_{n=-oo..+oo} (4^n + 1)^(2*n) / 4^(2*n^2+n).
A(1/5) = 1.4841513733060571811336245213703004776194631749017...
  where A(1/5) = Sum_{n=-oo..+oo} (5^n + 1)^(2*n) / 5^(2*n^2+n).
(End)

Paul D. Hanna, Table of n, a(n) for n = 0..8200

Cf. A260116, A260148, A217668, A260180, A260361.
Cf. A261605, A363558, A363559, A363569, A363561.
Cf. A378573.

with(numtheory):
seq(add(binomial(2*n/d, d-1) + (-1)^(n/d+1) * binomial(n/d, 2*d-1), d in divisors(n)), n = 1..70); # Peter Bala, Mar 02 2025

terms = 100; max = 2 terms; 1/2 Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *)

{a(n) = my(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k/2 + O(x^(2*n+2)) ); polcoef(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = my(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k /2  + O(x^(2*n+2)) ); polcoef(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = my(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k))/2 + O(x^(n+1)) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = my(A=1); A = sum(k=-n-1, n+1, x^k*(1+x^k)^(2*k) + O(x^(n+1)) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = my(A=1); A = sum(k=-n-1, n+1, x^(2*k^2-k)/(1-x^k + O(x^(n+1)))^(2*k)  ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))

The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n.
(2) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.
(3) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.
(4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.
(5) A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n).
(6) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^n)^(2*n).
(7) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - x^n)^(2*n).
(8) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + x^n)^(2*n).
a(2^n) = 1 for n > 0 (conjecture).
a(p) = p+1 for primes p > 3 (conjecture).
From Peter Bala, Jan 23 2021: (Start)
The following are conjectural:
A(x^2) = Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1) )^(2*n+1).
Equivalently: A(x^2) = Sum_{n = -oo..+oo} x^(4*n^2 + 2*n)/(1 + x^(2*n+1))^(2*n+1).
a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(4*n+2)
More generally, for k = 1,2,3,..., a((2^k)*(2*n + 1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(2^(k+1)*(2*n+1)).
a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2*n).
More generally, for k = 1,2,3,...,
a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2^(k+1)*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2^(k+1)*n).
a(4*n+2) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 + x^n)^(2*n) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 - x^n)^(2*n).
a(n) = [x^(2*n)] Sum_{n = -oo..+oo} (-1)^n*x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2*n+1).
For k = 1,2,3,...,
a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2^(k+1)*(2*n+1)).
(End)
From Peter Bala, Mar 02 2025: (Start)
a(n) = Sum_{d divides n} (binomial(2*n/d, d-1) + (-1)^(n/d+1) * binomial(n/d, 2*d-1)) for n >= 1.
Hence, a(p) = p + 1 for primes p > 3 and a(2^n) = 1 for n > 0 as conjectured above. (End)

A143862 Number of compositions of n such that every part is divisible by number of parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 9, 1, 7, 7, 9, 1, 19, 1, 14, 16, 11, 1, 43, 2, 13, 29, 34, 1, 56, 1, 51, 46, 17, 16, 130, 1, 19, 67, 139, 1, 105, 1, 142, 162, 23, 1, 315, 2, 151, 121, 246, 1, 219, 211, 321, 154, 29, 1, 1021, 1, 31, 219, 488, 496, 495, 1, 594, 232, 834, 1, 1439, 1

Offset: 0

Vladeta Jovovic, Sep 03 2008

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A143773, A261605.

{a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d-1,d-1) ))}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018

G.f.: Sum_{k>=0} x^(k^2) / (1 - x^k)^k.
G.f.: 1 + Sum_{n>=1} (1 + x^n)^(n-1) * x^n. - Paul D. Hanna, Jul 09 2019
a(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018
G.f.: 1 + Sum_{n>=1} (x^n/(1-x^n))^n (conjecture). - Joerg Arndt, Jan 04 2024
For prime p, a(p) = 1, a(2*p) = p and a(p^2) = 2. - Peter Bala, Mar 02 2025

More terms from Franklin T. Adams-Watters, Apr 09 2009

A261608 G.f.: Sum_{n=-oo..+oo, n<>0} x^(n^2) / (1 - x^n)^(n+1).

Original entry on oeis.org

2, 1, 4, 5, 6, 6, 8, 16, 12, 15, 12, 32, 14, 28, 32, 52, 18, 55, 20, 74, 72, 66, 24, 160, 28, 91, 140, 146, 30, 205, 32, 271, 244, 153, 72, 442, 38, 190, 392, 563, 42, 518, 44, 505, 788, 276, 48, 1510, 52, 451, 852, 896, 54, 1086, 728, 1748, 1180, 435, 60, 3291, 62, 496, 1648, 2867, 1848, 2101, 68, 2481, 2072, 1953, 72, 7634

Offset: 1

Paul D. Hanna, Aug 26 2015

nonn

			G.f.: A(x) = 2*x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 8*x^7 + 16*x^8 + 12*x^9 + 15*x^10 + 12*x^11 + 32*x^12 + 14*x^13 + 28*x^14 +...
where A(x) = N(x) + P(x) such that
N(x) = x*(x-1)^0 + x^2*(x^2-1) + x^3*(x^3-1)^2 + x^4*(x^4-1)^3 + x^5*(x^5-1)^4 + x^6*(x^6-1)^5 + x^7*(x^7-1)^6 + x^8*(x^8-1)^7 +...
P(x) = x/(1-x)^2 + x^4/(1-x^2)^3 + x^9/(1-x^3)^4 + x^16/(1-x^4)^5 + x^25/(1-x^5)^6 + x^36/(1-x^6)^7 + x^49/(1-x^7)^8 +...
explicitly,
N(x) = x - x^2 + x^3 + x^5 - 3*x^6 + x^7 + 2*x^8 + 2*x^9 - 5*x^10 + x^11 + x^12 + x^13 - 7*x^14 + 7*x^15 + 7*x^16 + x^17 - 19*x^18 + x^19 + 4*x^20 +...
P(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A261605.

{a(n) = polcoeff(sum(m=-n-1, n+1, if(m!=0, x^(m^2)/(1-x^m +x*O(x^n))^(m+1))), n)}
for(n=1, 60, print1(a(n), ", "))

{a(n) = polcoeff(sum(m=-n-1, n+1, if(m!=0, x^m*(x^m-1 +x*O(x^n))^(m-1))), n)}
for(n=1, 60, print1(a(n), ", "))

{a(n) = polcoeff(sum(m=-n-1, n+1, x^(m^2)/(1+x^m +x*O(x^n))^m), n)}
for(n=1, 60, print1(a(n), ", "))

{a(n) = polcoeff(sum(m=-n-1, n+1, (1 + x^m +x*O(x^n))^m), n)}
for(n=1, 60, print1(a(n), ", "))

G.f.: Sum_{n=-oo..+oo, n<>0} x^n * (x^n - 1)^(n-1).
G.f.: Sum_{n=-oo..+oo} x^(n^2)/(1 + x^n)^n, where the sum is taken to exclude the coefficient of x^0.
G.f.: Sum_{n=-oo..+oo} (1 + x^n)^n, where the sum is taken to exclude the coefficient of x^0.
G.f.: x * d/dx Sum_{n=-oo..+oo, n<>0} (1/n^2) * x^(n^2)/(1 - x^n)^n. - Paul D. Hanna, Nov 16 2017

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

Paul D. Hanna, Nov 21 2018

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.

			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 + ...

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

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

{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(""))

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 ).

cp's OEIS Frontend

A260147 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n).

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A143862 Number of compositions of n such that every part is divisible by number of parts.

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A261608 G.f.: Sum_{n=-oo..+oo, n<>0} x^(n^2) / (1 - x^n)^(n+1).

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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.

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