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.

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A361892 a(n) = S(7,2*n-1)/S(1,2*n-1), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 43, 9451, 6031627, 6571985126, 9140730357409, 14801600281919487, 26927918031565051915, 53804800109969394477580, 116002825041515533807200418, 266118189111094898593879923346, 642598035707739308769581970619393
Offset: 1

Views

Author

Peter Bala, Mar 30 2023

Keywords

Comments

Odd bisection of A361891.
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 5.

Links

Crossrefs

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A382394 ( S(3,2*n-1) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361887 ( S(5,n) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361891 ( S(7,n)/S(1,n) ).

Programs

  • Maple
    seq(add( ( binomial(2*n-1,k) - binomial(2*n-1,k-1) )^7/binomial(2*n-1,n-1), k = 0..n-1), n = 1..20);
  • Mathematica
    Table[Sum[(Binomial[2*n-1, k]-Binomial[2*n-1, k-1])^7 / Binomial[2*n-1, n-1], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 24 2025 *)
  • Python
    from math import comb
def A361892(n): return sum((comb((n<<1)-1,j)*(m:=n-j<<1)//(m+j))**7 for j in range(n))//comb((n<<1)-1,n-1) # Chai Wah Wu, Mar 25 2025

Formula

a(n) = 1/binomial(2*n-1,n-1) * Sum_{k = 0..n-1} ( (2*n - 2*k)/(2*n - k) * binomial(2*n-1,k) )^7 for n >= 1.
a(n) ~ 3 * 2^(12*n+1) / (2401 * Pi^3 * n^6). - Vaclav Kotesovec, Mar 24 2025

Extensions

Offset changed to 1 by Georg Fischer, Nov 20 2024

A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

Original entry on oeis.org

1, 1, 9, 190, 5705, 204876, 8209278, 354331692, 16140234825, 765868074400, 37525317999884, 1886768082651816, 96906387191038334, 5066711735118128200, 268954195756648761900, 14464077426547576156440, 786729115199980286001225, 43219452658242723841261800
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2025

Keywords

Comments

Let b_k(n) = Sum_{j=0..n} A128899(n,j)^k. b_1(n) = binomial(2*n-1,n) = A088218(n) and b_2(n) = A024492(n-1) for n > 0.

Links

Crossrefs

Cf. A024492, A088218.
Cf. A001700, A003161, A128899, A183069.

Programs

  • PARI
    a128899(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2);
a(n) = sum(k=0, n, a128899(n, k)^3);

Formula

a(n) = binomial(2*n,n)/2 * A183069(n) for n > 0.
a(n) = A003161(2*n-1) for n > 0.

A129130 Number of triples of standard tableaux with the same shape of height less than or equal to three.

Original entry on oeis.org

1, 1, 2, 10, 63, 531, 6201, 70477, 897149, 12772405, 188334604, 2939523104, 47902337803, 809518276503, 14134544880444, 252955559204532, 4651455689358657, 87356706437180529, 1669767921758484702
Offset: 0

Views

Author

Mike Zabrocki, Mar 30 2007

Keywords

Examples

			f_111 = f_3 = 1, f_21 = 2 therefore a(3) = f_111^3 + f_21^3 + f_3^3 = 1+8+1 = 10

Links

Crossrefs

Cf. A001006, A005802, A003161.

Formula

sum_lambda f_lambda^3 where the sum is over lambda partitions of length less than or equal to 3 and f_lambda is the number of standard tableaux of shape lambda
Previous Showing 11-13 of 13 results.

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