cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A359670 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 21, 12, 1, 14, 62, 68, 20, 1, 24, 162, 284, 170, 30, 1, 40, 384, 998, 970, 360, 42, 1, 64, 855, 3092, 4410, 2720, 679, 56, 1, 100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1, 154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1, 232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

Related identity: 0 = Sum_{-oo..+oo} (-1)^n * x^n * (y + x^n)^n, which holds formally for all y.
T(n,0) = A015128(n), the number of overpartitions of n, for n >= 0.
T(n+1,1) = A022571(n), the coefficient of x^n in Product_{m>=1} (1 + x^m)^6, for n >= 0.
A359711(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A359712(n) = Sum_{k=0..n} T(n,k)*2^k for n >= 0.
A359713(n) = Sum_{k=0..n} T(n,k)*3^k for n >= 0.
A363104(n) = Sum_{k=0..n} T(n,k)*4^k for n >= 0.
A363105(n) = Sum_{k=0..n} T(n,k)*5^k for n >= 0.
A359714(n) = T(2*n,n) for n >= 0 (central terms).
A359715(n) = T(n+2,2) for n >= 0.
A359718(n) = T(n+3,3) for n >= 0.
A363142(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023
From Paul D. Hanna, May 20 2023: (Start)
A363182(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 2^(n-2*k) for n >= 0.
A363183(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 3^(n-2*k) for n >= 0.
A363184(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 4^(n-2*k) for n >= 0.
A363185(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 5^(n-2*k) for n >= 0. (End)

Examples

			G.f.: A(x,y) = 1 + x*(2 + y) + x^2*(4 + 6*y + y^2) + x^3*(8 + 21*y + 12*y^2 + y^3) + x^4*(14 + 62*y + 68*y^2 + 20*y^3 + y^4) + x^5*(24 + 162*y + 284*y^2 + 170*y^3 + 30*y^4 + y^5) + x^6*(40 + 384*y + 998*y^2 + 970*y^3 + 360*y^4 + 42*y^5 + y^6) + x^7*(64 + 855*y + 3092*y^2 + 4410*y^3 + 2720*y^4 + 679*y^5 + 56*y^6 + y^7) + x^8*(100 + 1806*y + 8724*y^2 + 17172*y^3 + 15627*y^4 + 6608*y^5 + 1176*y^6 + 72*y^7 + y^8) + x^9*(154 + 3648*y + 22904*y^2 + 59545*y^3 + 74682*y^4 + 47089*y^5 + 14392*y^6 + 1908*y^7 + 90*y^8 + y^9) + x^10*(232 + 7110*y + 56679*y^2 + 188700*y^3 + 311530*y^4 + 271698*y^5 + 125160*y^6 + 28764*y^7 + 2940*y^8 + 110*y^9 + y^10) + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 0, k = 0..n, begins
[1];
[2, 1];
[4, 6, 1];
[8, 21, 12, 1];
[14, 62, 68, 20, 1];
[24, 162, 284, 170, 30, 1];
[40, 384, 998, 970, 360, 42, 1];
[64, 855, 3092, 4410, 2720, 679, 56, 1];
[100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1];
[154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1];
[232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1];
[344, 13434, 133516, 556085, 1169100, 1342684, 860664, 300888, 53640, 4345, 132, 1];
[504, 24702, 301664, 1542640, 4029237, 5884160, 4980320, 2438712, 666240, 94490, 6204, 156, 1];
[728, 44361, 657368, 4065868, 12940766, 23411339, 25215416, 16367874, 6302148, 1377464, 158708, 8606, 182, 1];
[1040, 78006, 1387854, 10253720, 39153924, 85994062, 114672768, 94919382, 48660900, 15071628, 2687454, 256022, 11648, 210, 1]; ...
RELATED SERIES.
Given g.f. F(x) of A361770, where
F(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ... + A361770(n)*x^n + ...
then
(1) F(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * F(x)^k,
(2) F(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^2 + x^(n-1))^(n+1).
Given g.f. G(x) of A363135, where
G(x) = 1 + 3*x + 17*x^2 + 133*x^3 + 1201*x^4 + 11796*x^5 + 122192*x^6 + 1314266*x^7 + 14536760*x^8 + 164299909*x^9 + ... + A363135(n)*x^n + ...
then
(1) G(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * G(x)^(2*k),
(2) G(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^3 + x^(n-1))^(n+1).

Links

Crossrefs

Cf. A359711 (row sums), A359712 (y=2), A359713 (y=3), A363104(y=4), A363105 (y=5).
Cf. A359714 (central terms), A359715 (column 2), A359718 (column 3).
Cf. A363142, A363182, A363183, A363184, A363185.
Cf. A361770, A363135, A363136, A363137.
Cf. A359720, A293600.

Programs

  • PARI
    {T(n,k) = my(A=1); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( polcoeff( A,n,x),k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))
  • PARI
    {T(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); polcoeff( A[n+1],k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k may be described as follows.
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).
(2) x*y = Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^(n+1).
(3) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n-1).
(4) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^n ].
(5) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^(n+1))^n ].
From Paul D. Hanna, May 18 2023: (Start)
(6) y = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (y*A(x,y) + x^n)^n.
(7) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (y*A(x,y) + x^n)^n ].
(8) x*y = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (y*A(x,y) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^n. (End)

A361770 Expansion of g.f. A(x) satisfying A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 14, 80, 510, 3498, 25145, 186972, 1426159, 11096944, 87736474, 702837098, 5692337206, 46533458472, 383450469145, 3181746494524, 26562082580277, 222941953595054, 1880174585677589, 15924467403391355, 135396623401761765, 1155230973031795808, 9888061401816818319
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Comments

Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)).

Examples

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A) + x^2*(4 + 6*A + A^2) + x^3*(8 + 21*A + 12*A^2 + A^3) + x^4*(14 + 62*A + 68*A^2 + 20*A^3 + A^4) + x^5*(24 + 162*A + 284*A^2 + 170*A^3 + 30*A^4 + A^5) + x^6*(40 + 384*A + 998*A^2 + 970*A^3 + 360*A^4 + 42*A^5 + A^6) + x^7*(64 + 855*A + 3092*A^2 + 4410*A^3 + 2720*A^4 + 679*A^5 + 56*A^6 + A^7) + x^8*(100 + 1806*A + 8724*A^2 + 17172*A^3 + 15627*A^4 + 6608*A^5 + 1176*A^6 + 72*A^7 + A^8) + ... + x^n*(Sum_{k=0..n} A359670(n,k)*A(x)^k) + ...
Also, A(x) = G(x,y=1) where G(x,y) satisfies
y*G(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*G(x,y)^2 + x^(n-1))^(n+1).
Explicitly,
G(x,y) = 1 + x*(2 + y) + x^2*(4 + 8*y + 2*y^2) + x^3*(8 + 37*y + 30*y^2 + 5*y^3) + x^4*(14 + 136*y + 234*y^2 + 112*y^3 + 14*y^4) + x^5*(24 + 432*y + 1320*y^2 + 1260*y^3 + 420*y^4 + 42*y^5) + x^6*(40 + 1232*y + 6093*y^2 + 9824*y^3 + 6240*y^4 + 1584*y^5 + 132*y^6) + x^7*(64 + 3245*y + 24402*y^2 + 60543*y^3 + 62880*y^4 + 29403*y^5 + 6006*y^6 + 429*y^7) + x^8*(100 + 8024*y + 87754*y^2 + 315616*y^3 + 490405*y^4 + 365816*y^5 + 134134*y^6 + 22880*y^7 + 1430*y^8) + x^9*(154 + 18832*y + 289812*y^2 + 1448744*y^3 + 3178302*y^4 + 3476418*y^5 + 1993992*y^6 + 598312*y^7 + 87516*y^8 + 4862*y^9) + ...

Links

Crossrefs

Cf. A359670, A359711, A363135, A363136, A363137.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^2 + x^(n-1))^(n+1) )/Ser(A), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^2 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^k.
(2) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^2 + x^n)^n.
(4) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n-1).
(5) x*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^2 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^2*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^2 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^n.
a(n) ~ c * d^n / n^(3/2), where d = 9.156930044633747979075094492861543774480990540... and c = 0.74413616954012053890115400925213042708811... - Vaclav Kotesovec, Jul 03 2025

A363136 Expansion of g.f. A(x) satisfying A(x)^3 = Sum_{n=-oo..+oo} (-x)^n * (A(x)^4 + x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 20, 201, 2364, 30356, 412223, 5821790, 84640367, 1258323895, 19041449659, 292322012264, 4541588520144, 71272574697572, 1128153098245655, 17990251268286993, 288748431461195204, 4660994483707782316, 75619617951059214712, 1232402387922242020729
Offset: 0

Views

Author

Paul D. Hanna, May 26 2023

Keywords

Comments

Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)^3).

Examples

			G.f.: A(x) = 1 + 3*x + 20*x^2 + 201*x^3 + 2364*x^4 + 30356*x^5 + 412223*x^6 + 5821790*x^7 + 84640367*x^8 + 1258323895*x^9 + 19041449659*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A^3) + x^2*(4 + 6*A^3 + A^6) + x^3*(8 + 21*A^3 + 12*A^6 + A^9) + x^4*(14 + 62*A^3 + 68*A^6 + 20*A^9 + A^12) + x^5*(24 + 162*A^3 + 284*A^6 + 170*A^9 + 30*A^12 + A^15) + x^6*(40 + 384*A^3 + 998*A^6 + 970*A^9 + 360*A^12 + 42*A^15 + A^18) + x^7*(64 + 855*A^3 + 3092*A^6 + 4410*A^9 + 2720*A^12 + 679*A^15 + 56*A^18 + A^21) + x^8*(100 + 1806*A^3 + 8724*A^6 + 17172*A^9 + 15627*A^12 + 6608*A^15 + 1176*A^18 + 72*A^21 + A^24) + ... + x^n*(Sum_{k=0..n} A359670(n,k) * A(x)^(3*k)) + ...
RELATED SERIES.
A(x)^2 = 1 + 6*x + 49*x^2 + 522*x^3 + 6334*x^4 + 82936*x^5 + 1141543*x^6 + 16281486*x^7 + 238492002*x^8 + 3566598406*x^9 + ...
A(x)^3 = 1 + 9*x + 87*x^2 + 990*x^3 + 12450*x^4 + 166767*x^5 + 2332148*x^6 + 33654549*x^7 + 497426898*x^8 + 7492104096*x^9 + ...
A(x)^4 = 1 + 12*x + 134*x^2 + 1632*x^3 + 21333*x^4 + 293036*x^5 + 4171534*x^6 + 61001912*x^7 + 910937790*x^8 + 13833090604*x^9 + ...

Links

Crossrefs

Cf. A361770, A363135, A363137.
Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^4 + x^(n-1))^(n+1) )/Ser(A)^3, #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^4 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^(3*k).
(2) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^4 + x^(n-1))^(n+1).
(3) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^4 + x^n)^n.
(4) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^(n-1).
(5) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^4 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^4 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^4*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^4 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^n.

A363137 Expansion of g.f. A(x) satisfying A(x)^4 = Sum_{n=-oo..+oo} (-x)^n * (A(x)^5 + x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 23, 284, 4125, 65526, 1102403, 19305377, 348217156, 6425056149, 120700893495, 2300815588583, 44391646154596, 865243089927133, 17011581975085968, 336981451741477122, 6719019528496352690, 134742110298875293027, 2715909284023948643846, 54992586234084937679092
Offset: 0

Views

Author

Paul D. Hanna, May 26 2023

Keywords

Comments

Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)^4).

Examples

			G.f.: A(x) = 1 + 3*x + 23*x^2 + 284*x^3 + 4125*x^4 + 65526*x^5 + 1102403*x^6 + 19305377*x^7 + 348217156*x^8 + 6425056149*x^9 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A^4) + x^2*(4 + 6*A^4 + A^8) + x^3*(8 + 21*A^4 + 12*A^8 + A^12) + x^4*(14 + 62*A^4 + 68*A^8 + 20*A^12 + A^16) + x^5*(24 + 162*A^4 + 284*A^8 + 170*A^12 + 30*A^16 + A^20) + x^6*(40 + 384*A^4 + 998*A^8 + 970*A^12 + 360*A^16 + 42*A^20 + A^24) + x^7*(64 + 855*A^4 + 3092*A^8 + 4410*A^12 + 2720*A^16 + 679*A^20 + 56*A^24 + A^28) + x^8*(100 + 1806*A^4 + 8724*A^8 + 17172*A^12 + 15627*A^16 + 6608*A^20 + 1176*A^24 + 72*A^28 + A^32) + ... + x^n*(Sum_{k=0..n} A359670(n,k) * A(x)^(4*k)) + ...
RELATED SERIES.
A(x)^2 = 1 + 6*x + 55*x^2 + 706*x^3 + 10483*x^4 + 168866*x^5 + 2868368*x^6 + 50582368*x^7 + 917211505*x^8 + 16994216980*x^9 + ...
A(x)^3 = 1 + 9*x + 96*x^2 + 1293*x^3 + 19695*x^4 + 322449*x^5 + 5539013*x^6 + 98484537*x^7 + 1797074331*x^8 + 33461795117*x^9 + ...
A(x)^4 = 1 + 12*x + 146*x^2 + 2072*x^3 + 32463*x^4 + 541188*x^5 + 9414694*x^6 + 168962408*x^7 + 3105263987*x^8 + 58149612672*x^9 + ...
A(x)^5 = 1 + 15*x + 205*x^2 + 3070*x^3 + 49570*x^4 + 842723*x^5 + 14864320*x^6 + 269521315*x^7 + 4992898830*x^8 + 94091310230*x^9 + ...

Links

Crossrefs

Cf. A361770, A363135, A363136.
Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^5 + x^(n-1))^(n+1) )/Ser(A)^4, #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^5 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^(4*k).
(2) A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^5 + x^(n-1))^(n+1).
(3) A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^5 + x^n)^n.
(4) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^(n-1).
(5) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^5 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^5 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^5*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^5 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^n.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.