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 1-3 of 3 results.

A275935 Shifts 5 places left under binomial transform.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 225, 455, 1260, 4555, 17760, 67265, 241015, 818705, 2666400, 8464210, 26791045, 87104270, 300213875, 1119214050, 4500888827, 19104042345, 83376236115, 366831787085, 1609394914730, 7015234913278, 30426949154855, 131992116224295, 577090099245575, 2565792536742865, 11698401074992087, 55012217948708040
Offset: 0

Views

Author

Olivier Gérard, Aug 12 2016

Keywords

Crossrefs

Cf. A000998, A275934.

Formula

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+5).
G.f. A(x) satisfies: A(x) = x^4 + x^5 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

A275936 Shifts 6 places under binomial transform.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 810, 1464, 3262, 10269, 40404, 165635, 653580, 2439069, 8626470, 29121393, 94647798, 299273206, 933818700, 2935248294, 9557815564, 33225405312, 125646127446, 514785555355, 2232901148760, 9976014439674, 44944467146100, 201608952292578, 895062795448170
Offset: 0

Views

Author

Olivier Gérard, Aug 12 2016

Keywords

Links

Crossrefs

Cf. A000998, A275934, A275935.

Programs

  • Maple
    A:= Array(0..10000):
A[5]:= 1:
for n from 6 to 100 do
  A[n]:= add(binomial(n-6,i)*A[i],i=0..n-6);
od:
convert(A,list); # Robert Israel, Mar 04 2024

Formula

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+6).
G.f. A(x) satisfies: A(x) = x^5 + x^6 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 9, 27, 81, 245, 761, 2493, 8849, 34519, 147057, 670327, 3198561, 15732905, 79174929, 407127897, 2145061729, 11635963499, 65309080185, 380583443187, 2304629301041, 14475031232285, 93943897651017, 627220447621973, 4290783719133041, 29988917377046207
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

Comments

Shifts 4 places left under 2nd-order binomial transform.

Crossrefs

Cf. A004211, A007472, A010748, A210541, A275934, A351342, A351344, A351345.

Programs

  • Mathematica
    nmax = 29; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 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 < 4, 1, Sum[Binomial[n - 4, k] 2^k a[n - k - 4], {k, 0, n - 4}]]; Table[a[n], {n, 0, 29}]

Formula

a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * 2^k * a(n-k-4).
Showing 1-3 of 3 results.

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