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 1-6 of 6 results.

A355357 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 1, 4, 7, 20, 43, 110, 262, 674, 1684, 4397, 11320, 29938, 78641, 210044, 559724, 1507563, 4060585, 11016027, 29919220, 81673846, 223307300, 612851316, 1684816018, 4645243490, 12829177587, 35513736868, 98465916370, 273531234027, 760966444416
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..floor(n/2)} A355350(n-k,n-2*k) for n >= 0.
a(n) = A359720(n,0), for n >= 0.

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 20*x^5 + 43*x^6 + 110*x^7 + 262*x^8 + 674*x^9 + 1684*x^10 + 4397*x^11 + 11320*x^12 + ...
where
x = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 -+ ...
also,
x*P(x^2) = (1 - x^2*A(x))*(1 - 1/A(x)) * (1 - x^4*A(x))*(1 - x^2/A(x)) * (1 - x^6*A(x))*(1 - x^4/A(x)) * (1 - x^8*A(x))*(1 - x^6/A(x)) * ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355356.
Cf. A359720, A357221, A357222, A357223, A357224, A357225, A357226.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, Sqrt[-Log[r] * ((-1 + r) * QPochhammer[1/r, r^2] * (-2*Log[r] + (-1 + r)*(Log[1 - r^2] - Log[r - r^3]) + (-1 + r) * QPolyGamma[0, -1/2, r^2] - (-1 + r)*QPolyGamma[0, 1, r^2]) + 4*(-1 + r)^2 * r^2 * Log[r] * Derivative[0, 1][QPochhammer][1/r, r^2] + 2*r^3 * Log[r] * QPochhammer[1/r, r^2]^3 * Derivative[0, 1][QPochhammer][r^2, r^2]) / (Pi*r^2* QPochhammer[1/r, r^2] * (-4*r*Log[r]^2 + (-1 + r)^2 * QPolyGamma[1, -1/2, r^2]))]} /. FindRoot[ 1/QPochhammer[r^2] == (r*QPochhammer[1/r, r^2]^2)/(-1 + r)^2, {r, 1/3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)
  • PARI
    {a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(n+4));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) x*P(x^2) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (2) can be rewritten as the functional equation x/QPochhammer(x^2) = QPochhammer(y, x^2)/(1 - y) * QPochhammer(1/(x^2*y), x^2)/(1 - 1/(x^2*y)).
a(n) ~ c * d^n / n^(3/2), where d = 2.92005174190265697439941308343193651904071627244119127019370275824199... and c = 1.4709989760845501303394202030872391136773745007487301056274536584990... (End)

A357221 Coefficients in the power series A(x) such that: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 2, 8, 26, 97, 361, 1399, 5532, 22318, 91387, 379037, 1588769, 6720065, 28645624, 122937300, 530748439, 2303446566, 10043922651, 43979954296, 193309569331, 852599816069, 3772220833468, 16737583785420, 74461239372631, 332062396407641, 1484162266154404
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 97*x^5 + 361*x^6 + 1399*x^7 + 5532*x^8 + 22318*x^9 + 91387*x^10 + 379037*x^11 + 1588769*x^12 + ...
such that
x*A(x) = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A355361, A357222, A357223, A357224, A357225, A357226.

Programs

  • PARI
    {a(n,p=1) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n+1)), ceil(sqrt(n+1)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^2 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357222 Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 3, 15, 73, 391, 2180, 12620, 75056, 456004, 2817879, 17656517, 111919061, 716379379, 4623944175, 30062540989, 196692237527, 1294112710358, 8556766562091, 56829292404053, 378936456243142, 2535866861527016, 17025875430611442, 114654511539186113
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 73*x^4 + 391*x^5 + 2180*x^6 + 12620*x^7 + 75056*x^8 + 456004*x^9 + 2817879*x^10 + ...
such that
x*A(x)^2 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357223, A357224, A357225, A357226.

Programs

  • PARI
    {a(n,p=2) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^2 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^3 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357223 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 4, 25, 164, 1177, 8887, 69748, 563232, 4649672, 39063521, 332904462, 2870862974, 25005954906, 219675658337, 1944131038267, 17316793719372, 155122164103293, 1396584226654493, 12630315100857638, 114687815080027358, 1045218902425525155, 9557367319452886097
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 164*x^4 + 1177*x^5 + 8887*x^6 + 69748*x^7 + 563232*x^8 + 4649672*x^9 + 39063521*x^10 + ...
such that
x*A(x)^3 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357224, A357225, A357226.

Programs

  • PARI
    {a(n,p=3) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^3 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357224 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 38, 315, 2855, 27325, 272030, 2788042, 29221793, 311767823, 3374650902, 36968040004, 409076635878, 4565873250981, 51342245169913, 581093383193700, 6614534942714496, 75675364150733073, 869713202188274489, 10036085000519702155, 116238137830534589525
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 315*x^4 + 2855*x^5 + 27325*x^6 + 272030*x^7 + 2788042*x^8 + 29221793*x^9 + 311767823*x^10 + ...
such that
x*A(x)^4 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357223, A357225, A357226.

Programs

  • PARI
    {a(n,p=4) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^5 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357226 Coefficients in the power series A(x) such that: x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 7, 73, 861, 11112, 151822, 2159143, 31627140, 473909468, 7230035454, 111924733904, 1753728878625, 27759947012294, 443247756591472, 7130680715081049, 115466397372003479, 1880525144522628300, 30783524695736369568, 506215648672559259036, 8358521379108937920413
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151822*x^6 + 2159143*x^7 + 31627140*x^8 + 473909468*x^9 + 7230035454*x^10 + ...
such that
x*A(x)^6 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357223, A357224, A357225.

Programs

  • PARI
    {a(n,p=6) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^6 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^7 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
Showing 1-6 of 6 results.

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