A306859 -id:A306859 - cp's OEIS frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 4, 16, 1, 1, 1, 2, 8, 32, 1, 1, 1, 1, 5, 16, 64, 1, 1, 1, 1, 2, 11, 32, 128, 1, 1, 1, 1, 1, 6, 22, 64, 256, 1, 1, 1, 1, 1, 2, 16, 43, 128, 512, 1, 1, 1, 1, 1, 1, 7, 36, 85, 256, 1024, 1, 1, 1, 1, 1, 1, 2, 22, 72, 170, 512, 2048

Offset: 0

Seiichi Manyama, Mar 13 2019

			Square array begins:
     1,   1,  1,  1,  1,  1, 1, 1, 1, ...
     2,   1,  1,  1,  1,  1, 1, 1, 1, ...
     4,   2,  1,  1,  1,  1, 1, 1, 1, ...
     8,   4,  2,  1,  1,  1, 1, 1, 1, ...
    16,   8,  5,  2,  1,  1, 1, 1, 1, ...
    32,  16, 11,  6,  2,  1, 1, 1, 1, ...
    64,  32, 22, 16,  7,  2, 1, 1, 1, ...
   128,  64, 43, 36, 22,  8, 2, 1, 1, ...
   256, 128, 85, 72, 57, 29, 9, 2, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000079, A011782, A024493, A038503, A139398, A306847, A306852, A306859, A306860.

Cf. A306680.

T[n_, k_] := Sum[Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n,k*j).

A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^kbinomial(n,8k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, -8, -44, -164, -494, -1286, -3002, -6434, -12868, -24292, -43604, -74612, -121124, -183140, -245156, -245156, 0, 961376, 3749136, 10814896, 27293176, 63453016, 138975976, 290021896, 580043792, 1114351888, 2054677648, 3619173776

Offset: 0

Seiichi Manyama, Mar 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-2).

Column 8 of A307039.

Cf. A306859.

a[n_] := Sum[(-1)^k * Binomial[n,8*k], {k,0,Floor[n/8]}]; Array[a, 36, 0] (* Amiram Eldar, May 25 2021 *)

{a(n) = sum(k=0, n\8, (-1)^k*binomial(n, 8*k))}

N=66; x='x+O('x^N); Vec((1-x)^7/((1-x)^8+x^8))

G.f.: (1 - x)^7/((1 - x)^8 + x^8).

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - 2*a(n-8) for n > 7.

cp's OEIS Frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

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A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^kbinomial(n,8k).

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cp's OEIS Frontend

A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^k*binomial(n,8*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^kbinomial(n,8k).