A351660 -id:A351660 - cp's OEIS frontend

A351707 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - x)) / (1 - x)^2.

1, 1, 1, 1, 1, 3, 7, 15, 31, 65, 147, 373, 1051, 3157, 9761, 30573, 96965, 313999, 1049719, 3654303, 13284783, 50268837, 196638987, 789611161, 3238765671, 13540348965, 57710600953, 251163156089, 1118308871001, 5100825621147, 23838465463447, 114044805729151

Offset: 0

Ilya Gutkovskiy, Feb 16 2022

nonn

Cf. A040027, A210541, A351437, A351660.

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Binomial[n - 3, k + 1] a[k], {k, 0, n - 4}]]; Table[a[n], {n, 0, 31}]

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

A351754 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - x)) / (1 - x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 63, 129, 277, 651, 1703, 4859, 14581, 44711, 138053, 427709, 1334461, 4226501, 13724063, 46110643, 161210421, 586729441, 2213187623, 8591628435, 34081480017, 137398121611, 561199251633, 2320442726999, 9722362801575

Offset: 0

Ilya Gutkovskiy, Feb 18 2022

nonn

Cf. A040027, A210542, A351437, A351660, A351707, A351755.

nmax = 32; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 5, 1, Sum[Binomial[n - 4, k + 1] a[k], {k, 0, n - 5}]]; Table[a[n], {n, 0, 32}]

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

A351755 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - x)) / (1 - x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 63, 127, 257, 535, 1187, 2891, 7751, 22331, 66997, 204473, 626917, 1922395, 5899579, 18192715, 56739881, 180434023, 590010059, 1997588833, 7026454733, 25650892255, 96720885037, 374163527473, 1475021500693, 5893462132221

Offset: 0

Ilya Gutkovskiy, Feb 18 2022

nonn

Cf. A040027, A210543, A351437, A351660, A351707, A351754.

nmax = 34; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 6, 1, Sum[Binomial[n - 5, k + 1] a[k], {k, 0, n - 6}]]; Table[a[n], {n, 0, 34}]

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

cp's OEIS Frontend

A351707 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - x)) / (1 - x)^2.

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A351754 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - x)) / (1 - x)^2.

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A351755 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - x)) / (1 - x)^2.

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