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 1-3 of 3 results.

A363106 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).

Original entry on oeis.org

1, 2, 5, 14, 36, 98, 271, 752, 2124, 6052, 17375, 50292, 146469, 428992, 1262946, 3734748, 11089366, 33048498, 98819841, 296388284, 891436452, 2688029716, 8124678435, 24611028218, 74702698749, 227177047220, 692084278902, 2111883982538, 6454350205098, 19754469483978
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 36*x^4 + 98*x^5 + 271*x^6 + 752*x^7 + 2124*x^8 + 6052*x^9 + 17375*x^10 + 50292*x^11 + 146469*x^12 + ...

Links

Crossrefs

Cf. A359711, A363107, A363108, A363109, A363140.

Programs

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

Formula

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

A363107 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-2))^(n+1).

Original entry on oeis.org

1, 2, 6, 20, 60, 196, 644, 2128, 7178, 24374, 83496, 288420, 1002272, 3503748, 12311818, 43458316, 154038006, 548018604, 1956263020, 7004845080, 25153186956, 90554989440, 326790211458, 1181910952584, 4283416505940, 15553332981066, 56575492155764, 206136324338908
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 60*x^4 + 196*x^5 + 644*x^6 + 2128*x^7 + 7178*x^8 + 24374*x^9 + 83496*x^10 + 288420*x^11 + 1002272*x^12 + ...

Links

Crossrefs

Cf. A359712, A363106, A363108, A363109.

Programs

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

Formula

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

A363108 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^(n+1).

Original entry on oeis.org

1, 2, 7, 26, 86, 318, 1165, 4312, 16318, 62020, 238165, 921980, 3590145, 14067188, 55399442, 219172028, 870736366, 3472155062, 13892694747, 55759406580, 224427809830, 905659181212, 3663475842865, 14851965523630, 60334690089827, 245572722474460, 1001306332164918
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 26*x^3 + 86*x^4 + 318*x^5 + 1165*x^6 + 4312*x^7 + 16318*x^8 + 62020*x^9 + 238165*x^10 + 921980*x^11 + 3590145*x^12 + ...

Links

Crossrefs

Cf. A359713, A363106, A363107, A363109.

Programs

  • PARI
    {a(n) = my(A=[1], y=3); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(y - sum(n=-#A, #A, (-1)^n * x^(2*n) * (y*Ser(A) + x^(n-2))^(n+1) )/y, #A-1, x) ); A[n+1]}
for(n=0, 30, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=1, y=3); for(i=1, n,
A = 1/sum(m=-n,n, (-1)^m * x^(2*m) * (y*A + x^(m-2) + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 30, print1( a(n), ", "))

Formula

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

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