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 1-5 of 5 results.

A344490 a(n) = 1 + Sum_{k=0..n-3} binomial(n-2,k) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 20, 57, 171, 548, 1894, 6998, 27368, 112653, 486645, 2201162, 10397944, 51161168, 261571460, 1386846249, 7612315023, 43190917004, 252951090586, 1527112817054, 9492126182336, 60677428545165, 398489257136529, 2686088269505042, 18567557376240748
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2021

Keywords

Crossrefs

Cf. A000629, A210540, A344489, A344491, A344492, A344493.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k] , {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = (1 + x^2 A[x/(1 - x)])/((1 - x) (1 + x^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

A344491 a(n) = 1 + Sum_{k=0..n-4} binomial(n-3,k) * a(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 37, 97, 275, 810, 2468, 7840, 26182, 92047, 339029, 1299185, 5152244, 21091816, 89087652, 388318264, 1746324563, 8094422821, 38608318847, 189179752492, 950930369320, 4898477508796, 25841317224002, 139534769647745, 770795537345237, 4353368099507329
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2021

Keywords

Crossrefs

Cf. A000629, A210541, A344489, A344490, A344492, A344493.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 3, k] a[k] , {k, 0, n - 4}]; Table[a[n], {n, 0, 31}]
nmax = 31; A[] = 0; Do[A[x] = (1 + x^3 A[x/(1 - x)])/((1 - x) (1 + x^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = (1 + x^3 * A(x/(1 - x))) / ((1 - x) * (1 + x^3)).

A344492 a(n) = 1 + Sum_{k=0..n-5} binomial(n-4,k) * a(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 70, 170, 452, 1277, 3731, 11145, 34031, 106888, 348016, 1180538, 4173726, 15320402, 58053312, 225891952, 899492200, 3660479037, 15228099789, 64831944993, 282763031581, 1263953233142, 5788015999020, 27121892020940, 129849269955372, 634208223729772
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2021

Keywords

Crossrefs

Cf. A000629, A210542, A344489, A344490, A344491, A344493.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 4, k] a[k] , {k, 0, n - 5}]; Table[a[n], {n, 0, 33}]
nmax = 33; A[] = 0; Do[A[x] = (1 + x^4 A[x/(1 - x)])/((1 - x) (1 + x^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x/(1 - x))) / ((1 - x) * (1 + x^4)).

A344493 a(n) = 1 + Sum_{k=0..n-6} binomial(n-5,k) * a(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 135, 308, 767, 2059, 5821, 16963, 50312, 151189, 460981, 1433634, 4578748, 15110212, 51704075, 183423444, 672385222, 2534056116, 9768179743, 38357842713, 153070136072, 620275332697, 2553688944713, 10696223834397, 45654239302087
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2021

Keywords

Crossrefs

Cf. A000629, A210543, A344489, A344490, A344491, A344492.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 5, k] a[k] , {k, 0, n - 6}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 0; Do[A[x] = (1 + x^5 A[x/(1 - x)])/((1 - x) (1 + x^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

A346771 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)) / (1 - x^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 81, 325, 1429, 6851, 35443, 196507, 1160633, 7266561, 48022313, 333776331, 2432140759, 18528143535, 147201596073, 1216952016245, 10448532393869, 92999784076875, 856739848236627, 8156691628658019, 80147320081510673, 811770418508099905
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Links

Crossrefs

Cf. A000110, A000296, A040027, A060719, A186021, A344489.

Programs

  • Mathematica
    nmax = 25; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)]/(1 - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 25; CoefficientList[Series[Exp[-x] (2 Exp[Exp[x] - 1] - 1), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(-x) * (2 * exp(exp(x) - 1) - 1).
a(0) = a(1) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A186021(k).
a(n) = 2 * A000296(n) - (-1)^n.
Showing 1-5 of 5 results.

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