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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A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 8, 8, 2, 1, 6, 16, 32, 22, 2, 1, 7, 32, 128, 170, 72, 2, 1, 8, 64, 512, 1366, 992, 254, 2, 1, 9, 128, 2048, 10922, 16512, 6008, 926, 2, 1, 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1, 11, 512, 32768, 699050, 4196352, 6643782, 2973350, 232562, 12872, 2, 1
Offset: 0

Views

Author

Peter Luschny, Mar 18 2023

Keywords

Comments

Because of the interchangeability of 0 and 1 in the definition, A(n, k) is even if n, k >= 1. In other words, if the binary representation of a permutation of the defined type is counted, then so is the 1's complement of that representation.

Examples

			Array A(n, k) starts:
 [0] 1, 2,    3,      4,        5,          6,            7, ...  A000027
 [1] 1, 2,    4,      8,       16,         32,           64, ...  A000079
 [2] 1, 2,    8,     32,      128,        512,         2048, ...  A081294
 [3] 1, 2,   22,    170,     1366,      10922,        87382, ...  A007613
 [4] 1, 2,   72,    992,    16512,     261632,      4196352, ...  A070775
 [5] 1, 2,  254,   6008,   215766,    6643782,    215492564, ...  A070782
 [6] 1, 2,  926,  37130,  2973350,  174174002,  11582386286, ...  A070967
 [7] 1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, ...  A094211
.
Triangle T(n, k) starts:
 [0]  1;
 [1]  2,   1;
 [2]  3,   2,    1;
 [3]  4,   4,    2,     1;
 [4]  5,   8,    8,     2,      1;
 [5]  6,  16,   32,    22,      2,      1;
 [6]  7,  32,  128,   170,     72,      2,     1;
 [7]  8,  64,  512,  1366,    992,    254,     2,    1;
 [8]  9, 128, 2048, 10922,  16512,   6008,   926,    2, 1;
 [9] 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1;
.
A(2, 2) = 8 = card(0000, 1100, 1010, 1001, 0110, 0101, 0011, 1111).
A(1, 3) = 8 = card(000, 100, 010, 001, 110, 101, 011, 111).

Crossrefs

Rows: A000027 (n=0), A000079 (n=1), A081294 (n=2), A007613 (n=3), A070775 (n=4), A070782 (n=5), A070967 (n=6), A094211 (n=7), A070832 (n=8), A094213 (n=9), A070833 (n=10).
Variant: A308500 (upwards antidiagonals).
Cf. A167009 (main diagonal).

Programs

  • Maple
    T := (n, k) -> add(binomial((n - k)*k, j*k), j = 0 .. n-k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);
  • SageMath
    # In Python use this import:
# from sympy.utilities.iterables import multiset_permutations
def A(n: int, k: int) -> int:
    if n == 0: return k + 1
    count = 0
    for a in range(0, n * k + 1, n):
        S = [i < a for i in range(n * k)]
        count += Permutations(S).cardinality()
    return count
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size)]
for n in range(6): print(ARow(n, 5))

Formula

A(n, k) = Sum_{j=0..k} binomial(n*k, n*j).
T(n, k) = Sum_{j=0..n-k} binomial((n - k)*k, j*k).

A216358 G.f.: 1/( (1-32*x)*(1+11*x-x^2)^2 )^(1/5).

Original entry on oeis.org

1, 2, 129, 2258, 66266, 1711282, 48405689, 1366932878, 39516211006, 1152710434262, 33978897474149, 1008971023405798, 30155867955237721, 906105094582017192, 27351768342997448884, 828919276503075367768, 25208280600556937464286, 768948732346237772809572
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 129*x^2 + 2258*x^3 + 66266*x^4 + 1711282*x^5 +...
where 1/A(x)^5 = 1 - 10*x - 585*x^2 - 3830*x^3 + 705*x^4 - 32*x^5.
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 254*x^2/2 + 6008*x^3/3 + 215766*x^4/4 + 6643782*x^5/5 + 215492564*x^6/6 +...+ A070782(n)*x^n/n +...

Crossrefs

Cf. A216316, A216357, A070782.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(5*m, 5*j))*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A070782(n)*x^n/n ) where A070782(n) = Sum_{k=0..n} binomial(5*n,5*k).
a(n) ~ 2^(5*n+3) * ((25-11*sqrt(5))/2)^(1/10) * GAMMA(4/5) / (5 * 11^(2/5) * n^(4/5) * Pi). - Vaclav Kotesovec, Jul 31 2014

A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(k*n,k*j).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 8, 8, 5, 1, 2, 22, 32, 16, 6, 1, 2, 72, 170, 128, 32, 7, 1, 2, 254, 992, 1366, 512, 64, 8, 1, 2, 926, 6008, 16512, 10922, 2048, 128, 9, 1, 2, 3434, 37130, 215766, 261632, 87382, 8192, 256, 10, 1, 2, 12872, 232562, 2973350, 6643782, 4196352, 699050, 32768, 512, 11
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,     1,       1,         1, ...
   2,  2,    2,     2,       2,         2, ...
   3,  4,    8,    22,      72,       254, ...
   4,  8,   32,   170,     992,      6008, ...
   5, 16,  128,  1366,   16512,    215766, ...
   6, 32,  512, 10922,  261632,   6643782, ...
   7, 64, 2048, 87382, 4196352, 215492564, ...

Links

Crossrefs

Columns k=0..10 give A000027(n+1), A000079, A081294, A007613, A070775, A070782, A070967, A094211, A070832, A094213, A070833.
Main diagonal gives A167009.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[k*n, k*j], {j, 0, n}] ; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A177809 Symmetrical sequence:Binomial(n,5*m).

Original entry on oeis.org

1, 1, 1, 1, 252, 1, 1, 3003, 3003, 1, 1, 15504, 184756, 15504, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 142506, 30045015, 155117520, 30045015, 142506, 1, 1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1, 1, 658008, 847660528, 40225345056
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are A070782.
5th in the sequence of sequence Binomial(n,k*m),k=1,2,3,4,5,...

Examples

			{1},
{1, 1},
{1, 252, 1},
{1, 3003, 3003, 1},
{1, 15504, 184756, 15504, 1},
{1, 53130, 3268760, 3268760, 53130, 1},
{1, 142506, 30045015, 155117520, 30045015, 142506, 1},
{1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1},
{1, 658008, 847660528, 40225345056, 137846528820, 40225345056, 847660528, 658008, 1},
{1, 1221759, 3190187286, 344867425584, 3169870830126, 3169870830126, 344867425584, 3190187286, 1221759, 1},
{1, 2118760, 10272278170, 2250829575120, 47129212243960, 126410606437752, 47129212243960, 2250829575120, 10272278170, 2118760, 1}

Crossrefs

Cf. A086645, A034839, A070775, A177808, A139459, A070782.

Programs

  • Mathematica
    t[n_, m_] = Binomial[n, 5*m];
Table[Table[t[n, m], {m, 0, Floor[n/5]}], {n, 0, 50, 5}];
Flatten[%]

A177810 Triangle binomial(6*n,6*m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 924, 1, 1, 18564, 18564, 1, 1, 134596, 2704156, 134596, 1, 1, 593775, 86493225, 86493225, 593775, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 5245786, 11058116888, 353697121050, 353697121050, 11058116888, 5245786, 1
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are A070967. k=6 in binomial(k*n,k*m) sequence similar to k=2 in A086645, k=4 in A070775,...

Examples

			1;
1, 1;
1, 924, 1;
1, 18564, 18564, 1;
1, 134596, 2704156, 134596, 1;
1, 593775, 86493225, 86493225, 593775, 1;

Crossrefs

Cf. A034839, A177808, A139459, A070782, A177809, A070967.

Programs

  • Mathematica
    t[n_, m_] := Binomial[n, 6*m]; Flatten@Table[Table[t[n, m], {m, 0, n/6}], {n, 0, 42, 6}]

Formula

Left-right symmetric: binomial(6*n,6*m) = binomial(6*n,6*(n-m)).
Previous Showing 11-15 of 15 results.

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