A352865 -id:A352865 - cp's OEIS frontend

A352864 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k,k) * a(n-2*k-1).

1, 1, 1, 3, 6, 13, 34, 84, 230, 653, 1893, 5794, 18080, 58345, 193761, 657959, 2295398, 8177305, 29775086, 110676222, 419169483, 1617868052, 6353518921, 25376986471, 103017630200, 424704411564, 1777458163195, 7546547411488, 32490058003914, 141774055915497, 626739661952337

Offset: 0

Ilya Gutkovskiy, Apr 06 2022

nonn

Cf. A040027, A172383, A352865.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - k, k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x^2)]/(1 - x^2)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]

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

A360900 G.f. satisfies A(x) = 1 + x/(1 + x^3)^2 * A(x/(1 + x^3)).

Original entry on oeis.org

1, 1, 1, 1, -1, -4, -8, -10, 2, 40, 115, 174, 22, -736, -2495, -4276, -920, 20593, 75011, 135814, 17524, -788871, -2909061, -5248648, 623274, 38581252, 138036877, 235567666, -134440249, -2284840691, -7698637124, -11745925435, 15869626983, 157479613343

Offset: 0

Seiichi Manyama, Feb 25 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A352865, A360901.

Cf. A360892.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\3, (-1)^j*binomial(i-2*j, j)*v[i-3*j])); v;

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} (-1)^k * binomial(n-2*k,k) * a(n-1-3*k).

A360901 G.f. satisfies A(x) = 1 + x/(1 + x^4)^2 * A(x/(1 + x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -1, -4, -8, -13, -16, -3, 39, 126, 273, 411, 283, -619, -3149, -8201, -14496, -14368, 12984, 109920, 329552, 655436, 785583, -312525, -5009034, -16789151, -36433571, -48990090, 6965306, 286109557, 1044072824, 2414264953, 3490255449, -41486331

Offset: 0

Seiichi Manyama, Feb 25 2023

sign

Cf. A352865, A360900.

Cf. A360893.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\4, (-1)^j*binomial(i-3*j, j)*v[i-4*j])); v;

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

cp's OEIS Frontend

A352864 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k,k) * a(n-2*k-1).

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A360900 G.f. satisfies A(x) = 1 + x/(1 + x^3)^2 * A(x/(1 + x^3)).

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A360901 G.f. satisfies A(x) = 1 + x/(1 + x^4)^2 * A(x/(1 + x^4)).

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PARI

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