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.

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

Original entry on oeis.org

1, 0, 1, 6, 37, 240, 1693, 13446, 122329, 1261104, 14332681, 175123446, 2267871517, 30981705984, 446571784261, 6798161166486, 109220619908593, 1846729159654560, 32726973173941585, 605358657750562470, 11648701234354836565, 232655173657593759312
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

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

Crossrefs

Cf. A000994, A005012, A351057, A351143, A351144, A351150, A351151.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 6 x)]/(1 - 6 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] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 21}]

Formula

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

A351056 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, 6, 36, 221, 1431, 10121, 80311, 718106, 7111976, 76201501, 868288401, 10438492181, 132166853861, 1763179150946, 24776241643056, 365971430085021, 5662954240306111, 91450179009971181, 1536249848608545451, 26782376261726525126, 483792982362049317676
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

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

Crossrefs

Cf. A005011, A007472, A007476, A351049, A351050, A351057.

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).

A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

Original entry on oeis.org

0, 1, 0, 1, 12, 109, 900, 7309, 62280, 590185, 6402360, 78347593, 1042633908, 14648616757, 214421295132, 3266839420021, 52041902492496, 870810496011793, 15326196662766384, 283049655668743249, 5460180803581446684, 109489002283248831037, 2273856664328893182324
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

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

Crossrefs

Cf. A000995, A005012, A351028, A351053, A351057, A351128, A351132, A351152.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 6 x)]/(1 - 6 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] 6^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) * 6^k * a(n-k-2).

A351187 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, -5, 25, -131, 793, -6137, 60049, -670919, 7930321, -96775853, 1225237609, -16333089227, 232150489129, -3531321746465, 57178717416097, -975918663642767, 17400776511175201, -322309002081819221, 6188520430773389881, -123166171374344928275, 2542231599282355411897
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

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

Crossrefs

Cf. A010739, A051139, A318181, A350456, A351057, A351184, A351185, A351186.

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).
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.