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 1-4 of 4 results.

A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1
Offset: 0

Views

Author

Seiichi Manyama, May 19 2019

Keywords

Comments

For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020

Examples

			For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + 2*x + 4*x^2/2 + 8*x^3/6 + 14*x^4/24 + 20*x^5/120 + 20*x^6/720. So A(3,2) = 1 + 2 + 4 + 8 + 14 + 20 + 20 = 69.
Square array begins:
   1, 1,    1,        1,             1,                   1, ...
   1, 2,    5,       16,            65,                 326, ...
   1, 3,   19,      271,          7365,              326011, ...
   1, 4,   69,     5248,       1107697,           492911196, ...
   1, 5,  251,   110251,     191448941,        904434761801, ...
   1, 6,  923,  2435200,   35899051101,    1856296498826906, ...
   1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.
Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.
Main diagonal gives A274762.
Cf. A144510.

Formula

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * x^i/i! = (Sum_{i=0..n} x^i/i!)^k.

A308296 a(n) = (1/n!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_n=1..n} multinomial(i_1+i_2+...+i_n; i_1, i_2, ... , i_n).

Original entry on oeis.org

1, 1, 7, 842, 7958726, 15467641899285, 10893033763705794846727, 4247805448772073978048752721163278, 1299618941291522676629215597535104557826094801396, 419715170056359079715862408734598208208707081189266290220651371206
Offset: 0

Views

Author

Seiichi Manyama, May 19 2019

Keywords

Examples

			a(2) = (1/2) * (binomial(1+1,1) + binomial(1+2,2) + binomial(2+1,1) + binomial(2+2,2)) = 7.

Links

Crossrefs

Cf. A144510, A144512, A274762, A281901.

Programs

  • PARI
    {a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, x^j/j!)^n, i))/n!}

Formula

a(n) = A144510(n,n).

A234574 T(n,k) is the number of size k ordered submultisets of the regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 1 <= i <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680, 1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000, 1, 5, 25, 125, 625, 3125, 15620, 77980, 388220, 1923180, 9454620
Offset: 0

Views

Author

Thomas Wieder, Dec 29 2013

Keywords

Comments

A181567 gives the case for unordered submultisets.

Examples

			For n=2 we have the regular multiset L = [1,1,2,2].
We get the following ordered submultisets from L:
For k=0 1 multiset: []
For k=1 2 multisets: [1], [2]
For k=2 4 multisets: [1,1], [1,2], [2,1], [2,2]
For k=3 6 multisets: [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1]
For k=4 6 multisets: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
Triangle begins with:
  1;
  1, 1;
  1, 2, 4, 6, 6;
  1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680;
  1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000;
  ...

Links

Crossrefs

Cf. A181567.
Row sums give A274762.

Programs

  • Maple
    # first Maple program: see link above
# second Maple program:
b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..n)))
    end:
T:= (n, k)-> b(n, k, n)*k!:
seq(seq(T(n, k), k=0..n^2), n=0..5); # Alois P. Heinz, Jul 04 2016

Extensions

More terms from Alois P. Heinz, Jul 04 2016

A308323 a(n) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_n=0..n} (-1)^(i_1 + i_2 + ... + i_n) * multinomial(i_1 + i_2 + ... + i_n; i_1, i_2, ..., i_n).

Original entry on oeis.org

1, 0, 3, -692, 25413205, -247578134832564, 1049539393188856278390451, -2871690441592604257533055860715129272, 7040796701370550878156702345258827433898638874681737, -20490586759699718697495728993626502030472234718733294544426997558792280
Offset: 0

Views

Author

Seiichi Manyama, May 20 2019

Keywords

Links

Crossrefs

Main diagonal of A308322.
Cf. A274762.

Programs

  • PARI
    {a(n) = sum(i=0, n^2, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^n, i))}
Showing 1-4 of 4 results.

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