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.

A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

1, 0, 1, 5, 26, 145, 901, 6420, 52501, 480955, 4795626, 51066375, 576182001, 6879462680, 86955722401, 1162559359745, 16392133866026, 242734091500445, 3758825675820501, 60660434188558780, 1017770666417312501, 17725289455315892375, 320047193447632729626
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A000994, A005011, A351056, A351143, A351144, A351150, A351152.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

Original entry on oeis.org

1, 1, 1, 7, 49, 349, 2593, 20755, 184609, 1851289, 20735041, 253471039, 3310505425, 45630322741, 660993079393, 10065000586507, 161262522401089, 2717539655666353, 48053169836707969, 888408313419305719, 17108882037936283249, 342144175940842590349, 7089944927940141776545
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

Shifts 2 places left under 6th-order binomial transform.

Crossrefs

Cf. A005012, A007472, A007476, A351049, A351050, A351056.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

A351132 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

0, 1, 0, 1, 10, 76, 530, 3701, 27810, 237151, 2316350, 25135126, 292106400, 3559029501, 45211131460, 600619791201, 8384107777030, 123237338584576, 1904128564485610, 30789744821412401, 518479182191232950, 9057086806410632751, 163745788914416588050
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A000995, A005011, A351028, A351053, A351056, A351128, A351151, A351161.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5*x)) / (1 + 5*x).

Original entry on oeis.org

1, 1, 1, -4, 16, -69, 371, -2719, 24691, -243804, 2479276, -25931249, 284075601, -3320433179, 41744590941, -561939568544, 8008026088996, -119496752915869, 1854697111334891, -29870689367146379, 499291484226079551, -8668202648905259624, 156301404533216141576
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

Shifts 2 places left under 5th-order inverse binomial transform.

Crossrefs

Cf. A010739, A051139, A318180, A350456, A351056, A351184, A351185, A351187.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 5 x)]/(1 + 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-5)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-5)^k * a(n-k-2).
Showing 1-4 of 4 results.

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