A275936 -id:A275936 - cp's OEIS frontend

A275934 Shifts 4 places left under binomial transform.

0, 0, 0, 1, 0, 0, 0, 1, 4, 10, 20, 36, 68, 166, 540, 1961, 7012, 23878, 78004, 250311, 815196, 2787806, 10232556, 40266382, 166608080, 708407020, 3046352440, 13161936881, 57188405288, 251328286460, 1125890398160, 5177570523461, 24539362719532, 119861818560962, 601299401594540, 3082695751138656, 16075855888601716, 85005009812011810, 455172001509369028, 2468935975119176601, 13584735197391443020

Offset: 0

Olivier Gérard, Aug 12 2016

Cf. A000998, A275935, A275936.

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+4).

G.f. A(x) satisfies: A(x) = x^3 + x^4 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 9, 27, 81, 243, 729, 2189, 6601, 20141, 63009, 205989, 718905, 2720543, 11183601, 49321367, 228895201, 1097860903, 5371546897, 26598018425, 132755261681, 667027581401, 3376011676481, 17249045903945, 89270689572497, 470069622480667

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

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

Cf. A004211, A007472, A010750, A210543, A275936, A351342, A351343, A351344

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 6, 1, Sum[Binomial[n - 6, k] 2^k a[n - k - 6], {k, 0, n - 6}]]; Table[a[n], {n, 0, 31}]

a(0) = ... = a(5) = 1; a(n) = Sum_{k=0..n-6} binomial(n-6,k) * 2^k * a(n-k-6).

cp's OEIS Frontend

A275934 Shifts 4 places left under binomial transform.

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A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2x)) / (1 - 2x).

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cp's OEIS Frontend

A275934 Shifts 4 places left under binomial transform.

Views

Author

Keywords

Crossrefs

Formula

A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2*x)) / (1 - 2*x).

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Mathematica

Formula

A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2x)) / (1 - 2x).