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.

Previous Showing 11-16 of 16 results.

A110166 Row sums of Riordan array A110165.

Original entry on oeis.org

1, 4, 18, 85, 410, 1999, 9807, 48304, 238570, 1180615, 5851253, 29033074, 144190943, 716652070, 3564079250, 17734184365, 88280673770, 439625873215, 2189988826125, 10912480440850, 54389237971285, 271142650382080
Offset: 0

Views

Author

Paul Barry, Jul 14 2005

Keywords

Comments

Number of 5-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012
From Peter Bala, Jan 09 2022: (Start)
Conjectures: for k >= 2, the number of k-ary words of length n such that the number of 1's <= the number of 0's is equal to the coefficient of x^n in the expansion of ( k*x + 1/(1 + x) )^n, and satisfies the recurrence u(0) = 1, u(1) = k-1 and n*u(n) = (k-2)*(2*n-1)*u(n-1) - k*(k-4)*(n-1)* u(n-2) + k^(n-1) for n >= 2.
For cases see A027306 (k = 2), A027914 (k = 3) and A032443 (k = 4). (End)

Links

Crossrefs

Row sums A110165. Cf. A026375, A032443, A246437, A242586.

Programs

  • Maple
    seq( (1/2)*(5^n + add(binomial(n,k)*binomial(2*k,k), k = 0..n)), n = 0..30); # Peter Bala, Jan 08 2022
  • Mathematica
    Table[Sum[Sum[Binomial[n,j]Binomial[2j,j+k],{j,0,n}],{k,0,n}],{n,0,25}] (* Harvey P. Dale, Dec 16 2011 *)

Formula

G.f.: (1/sqrt(1-6*x+5*x^2))/(1-(1-3*x-sqrt(1-6*x+5*x^2))/(2*x)).
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} C(n, j)*C(2*j, j+k).
Recurrence: n*a(n) = (11*n-8)*a(n-1) - 5*(7*n-10)*a(n-2) + 25*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 5^n/2*(1+sqrt(5)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Oct 18 2012
From Peter Bala, Jan 08 2022: (Start)
a(n) = (1/2)*(5^n + A026375(n)) = (1/2)*(5^n + Sum_{k = 0..n} binomial(n,k) *binomial(2*k,k)).
a(n) = (1/2)*(5^n)*(1 + Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(-1/5)^k).
a(n) = [x^n] ( 5*x + 1/(1 + x) )^n.
a(0) = 1, a(1) = 4 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) + 5^(n-1) for n >= 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
Binomial transform of A032443. (End)

A025191 a(n) = Sum_{k=0..n} T(n,k), where T is the array defined in A025177.

Original entry on oeis.org

1, 1, 4, 11, 33, 97, 288, 855, 2544, 7577, 22590, 67399, 201215, 601017, 1795966, 5368659, 16053417, 48015873, 143649102, 429842511, 1286452725, 3850770081, 11528245602, 34517105907, 103360732956, 309543786441, 927106804368, 2776994293355
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Conjectures: a(n) = A027914(n)-A027914(n-1) = (A081673(n)-A081673(n-1))/2.

Extensions

a(1) corrected by Jason Yuen, Aug 05 2024

A027915 a(n) = Sum_{0<=j<=i, 0<=i<=n} A027907(i, j).

Original entry on oeis.org

1, 3, 9, 26, 76, 223, 658, 1948, 5782, 17193, 51194, 152594, 455209, 1358841, 4058439, 12126696, 36248370, 108385917, 324172566, 969801726, 2901883611, 8684735577, 25995833145, 77824036620, 233012973051, 697745695923
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Partial sums of A027914.

Formula

G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2))/(1-x), where G(k)= 1 + x*(2+3*x)*(4*k+1)/(4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013

A092436 a(n) = 1/2 + (-1)^n*(1/2 - A010060(floor(n/2))).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1
Offset: 1

Views

Author

Benoit Cloitre, Mar 23 2004

Keywords

Comments

From Jeffrey Shallit, Mar 02 2022: (Start)
Also, the parity of the number of 2's in the bijective base-2 representation of n - 1; this is the base-2 representation using the digits {1,2} in place of {0,1}.
Also, solution of the equation a = 0 mu(a), where mu is the Thue-Morse morphism 0 -> 01, 1 -> 10. (End)

Links

Crossrefs

Cf. A027914, A010060.

Programs

  • Mathematica
    Flatten[ NestList[ Function[l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {1, 0}})]}], {0}, 6]] (* Robert G. Wilson v, May 19 2005 *)
  • Python
    def A092436(n): return n.bit_count()&1^1 # Chai Wah Wu, Mar 03 2023

Formula

a(n) = 1-A010060(n). - Chai Wah Wu, Mar 03 2023

A026080 Sum{T(n,k)}, k = 0,1,...,n, where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 4, 7, 22, 64, 191, 567, 1689, 5033, 15013, 44809, 133816, 399802, 1194949, 3572693, 10684758, 31962456, 95633229, 286193409, 856610214, 2564317356, 7677475521, 22988860305, 68843627049, 206183053485, 617563017927, 1849887488987
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A027914.

A141905 Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 24, 8, 1, 1, 25, 70, 40, 10, 1, 1, 36, 165, 160, 60, 12, 1, 1, 49, 336, 525, 280, 84, 14, 1, 1, 64, 616, 1456, 1120, 448, 112, 16, 1, 1, 81, 1044, 3528, 3906, 2016, 672, 144, 18, 1, 1, 100, 1665, 7680, 11970, 8064, 3360, 960, 180, 20, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Keywords

Comments

Original definition: A skew trinomial summed triangular sequence of coefficients: T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
It is obscure how the above formula is used for the region where the sum reaches k > n-m, which needs a definition of the factorials at negative integer argument. If we trust the author's Mma implementation, Mma throws in some magic renormalization to cover these arguments. If we define, properly, t(n, k) = Sum_{j=0..n-k} n!/((n-k-j)!*j!*k!), then we recover just A038207. - R. J. Mathar, Feb 07 2014
Let p(n, k, j) = n!/((n-k-j)!*j!*k!), for j<=n-k and 0<= k <=n and p(n, k, j) = 0, for j > n-k and 0<= k <=n. It seems that T(n, k) coincides with Sum_{j=0..k} p(n, k, j). - Luis Manuel Rivera Martínez, Mar 04 2014

Examples

			Triangle begins as:
[0]  1;
[1]  1,   1;
[2]  1,   4,    1;
[3]  1,   9,    6,    1;
[4]  1,  16,   24,    8,     1;
[5]  1,  25,   70,   40,    10,    1;
[6]  1,  36,  165,  160,    60,   12,    1;
[7]  1,  49,  336,  525,   280,   84,   14,   1;
[8]  1,  64,  616, 1456,  1120,  448,  112,  16,   1;
[9]  1,  81, 1044, 3528,  3906, 2016,  672, 144,  18,  1;

Links

Crossrefs

Row sums are A027914.
Cf. A038207, A052509.

Programs

  • Magma
    [Binomial(n,k)*(&+[Binomial(n-k,j): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 29 2021
  • Maple
    A052509 := proc(n, k) option remember: if k = 0 or k = n then 1 else A052509(n-1, k) + A052509(n-2, k-1) fi end: T := (n, k) -> binomial(n, k)*A052509(n, k): seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Nov 26 2021
  • Mathematica
    T[n_, k_]:= Sum[n!/((n-k-j)!*j!*k!), {j,0,k}];
Table[T[n, k], {n, 0, 10}, {k,0,n}] // Flatten
  • Sage
    flatten([[binomial(n,k)*sum(binomial(n-k,j) for j in (0..k)) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 29 2021

Formula

T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
G.f.: (2*x)/((3*x - 1)*sqrt(-4*x^2*y + x^2 - 2*x + 1) - 4*x^2*y + x^2 - 2*x +1). - Vladimir Kruchinin, Oct 05 2020
T(n, k) = binomial(n, k)*hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021

Extensions

Edited by G. C. Greubel, Mar 29 2021
New name by Peter Luschny, Nov 26 2021
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.