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 1-5 of 5 results.

A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 37, 31, 4, 1, 1, 266, 842, 121, 5, 1, 1, 2431, 45296, 18252, 456, 6, 1, 1, 27007, 4061871, 7958726, 405408, 1709, 7, 1, 1, 353522, 546809243, 7528988476, 1495388159, 9268549, 6427, 8, 1
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Dec 15 2008, Jan 30 2009

Keywords

Examples

			Array begins:
1, 1,    1,       1,            1,                 1,                       1, ...
1, 2,    7,      37,          266,              2431,                   27007, ...
1, 3,   31,     842,        45296,           4061871,               546809243, ...
1, 4,  121,   18252,      7958726,        7528988476,          13130817809439, ...
1, 5,  456,  405408,   1495388159,    15467641899285,      361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...

Links

Crossrefs

For the transposed array see A144512.
Rows include A001515, A144416, A144508, A144509.
Columns include A048775, A144511.
A(n+1,n) gives A281901.
A(n,n) gives A308296.
Cf. A308292.

Programs

  • Maple
    b := proc(n, i, k) local r;
option remember;
if n = i then 1;
elif i < n then 0;
elif n < 1 then 0;
else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
end if;
end proc;
T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;
# Peter Luschny, Apr 26 2011
A144510 := proc(n, k) local m;
add(m!*coeff(expand((exp(x)*GAMMA(n+1,x)/GAMMA(n+1)-1)^k),x,m),m=k..k*n)/k! end: for row from 1 to 6 do seq(A144510(row, col), col = 0..5) od;
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Formula

T(n,k) = (1/k!)*Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} multinomial(i_1+i_2+...+i_k; i_1, i_2, ..., i_k).
T(n,k) = (1/k!)*Sum_{m=k..k*n} m! [x^m](e^x Gamma(n+1,x)/Gamma(n+1)-1)^k. Here [x^m]f(x) is the coefficient of x^m in the series expansion of f(x). - Peter Luschny, Apr 26 2011

A274762 Number of sequences with up to n copies each of 1,2,...,n.

Original entry on oeis.org

1, 2, 19, 5248, 191448941, 1856296498826906, 7843008902239185171370147, 21408941228439913825832318523364743824, 52400635808473472283994952631626957015306076632624953, 152306240915343870544748050434914720360496623911547121447677238156864610
Offset: 0

Views

Author

Alois P. Heinz, Jul 04 2016

Keywords

Examples

			a(0) = 1: () = the empty sequence.
a(1) = 2: (), 1.
a(2) = 19: (), 1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1122, 1212, 1221, 2112, 2121, 2211.

Links

Crossrefs

Row sums of A234574.
Main diagonal of A308292.
Cf. A000312, A034841.

Programs

  • Maple
    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..min(k, n))))
    end:
a:= n-> add(b(n, k, n)*k!, k=0..n^2):
seq(a(n), n=0..10);
  • Mathematica
    Table[Sum[k!*SeriesCoefficient[Sum[x^j/j!, {j, 0, n}]^n, {x, 0, k}], {k, 0, n^2}], {n, 0, 10}] (* Vaclav Kotesovec, May 24 2020 *)
  • PARI
    {a(n) = sum(i=0, n^2, i!*polcoef(sum(j=0, n, x^j/j!)^n, i))} \\ Seiichi Manyama, May 19 2019

Formula

a(n) ~ exp(11/12) * n^(n^2 - n/2 + 1) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, May 24 2020

A308322 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} (-1)^(i_1 + i_2 + ... + i_k) * 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, 0, 1, 1, 1, 1, 1, 1, -2, 3, 0, 1, 1, 9, 37, 9, 1, 1, 1, -44, 997, -692, 31, 0, 1, 1, 265, 44121, 148041, 14371, 111, 1, 1, 1, -1854, 2882071, -66211704, 25413205, -315002, 407, 0, 1, 1, 14833, 260415373, 53414037505, 120965241901, 4744544613, 7156969, 1513, 1, 1
Offset: 0

Views

Author

Seiichi Manyama, May 20 2019

Keywords

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 = 9.
Square array begins:
   1, 1,   1,       1,            1,                  1, ...
   1, 0,   1,      -2,            9,                -44, ...
   1, 1,   3,      37,          997,              44121, ...
   1, 0,   9,    -692,       148041,          -66211704, ...
   1, 1,  31,   14371,     25413205,       120965241901, ...
   1, 0, 111, -315002,   4744544613,   -247578134832564, ...
   1, 1, 407, 7156969, 935728207597, 545591130328772081, ...

Links

Crossrefs

Columns k=0..5 give A000012, A059841, A120305, A307318, A307324, A308325.
Rows n=0..1 give A000012, A182386.
Main diagonal gives A308323.
Cf. A308292.

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.

A308294 a(n) = Sum_{i_1=0..3} Sum_{i_2=0..3} ... Sum_{i_n=0..3} multinomial(i_1+i_2+...+i_n; i_1, i_2, ... , i_n).

Original entry on oeis.org

1, 4, 69, 5248, 1107697, 492911196, 396643610629, 522506795651464, 1050188527130093313, 3055485688346936896372, 12353356560641179964896741, 67171925010307462937573055504, 478268992794023738033117638364209, 4360663458863998067849091605547380428
Offset: 0

Views

Author

Seiichi Manyama, May 19 2019

Keywords

Examples

			a(2) = binomial(0+0,0) + binomial(0+1,1) + binomial(0+2,2) + binomial(0+3,3) + binomial(1+0,0) + binomial(1+1,1) + binomial(1+2,2) + binomial(1+3,3) + binomial(2+0,0) + binomial(2+1,1) + binomial(2+2,2) + binomial(2+3,3) + binomial(3+0,0) + binomial(3+1,1) + binomial(3+2,2) + binomial(3+3,3) = 69.

Links

Crossrefs

Row n=3 of A308292.

Programs

  • Mathematica
    Table[Total[CoefficientList[Series[(1 + x + x^2/2 + x^3/6)^n, {x, 0, 3*n}], x]*Range[0, 3*n]!], {n, 0, 15}] (* Vaclav Kotesovec, May 24 2020 *)
  • PARI
    {a(n) = sum(i=0, 3*n, i!*polcoef(sum(j=0, 3, x^j/j!)^n, i))}

Formula

a(n) ~ sqrt(Pi) * 3^(2*n + 1/2) * n^(3*n + 1/2) / (2^(n - 1/2) * exp(3*n - 1)). - Vaclav Kotesovec, May 24 2020

A308295 a(n) = Sum_{i_1=0..4} Sum_{i_2=0..4} ... Sum_{i_n=0..4} multinomial(i_1 + i_2 + ... + i_n; i_1, i_2, ..., i_n).

Original entry on oeis.org

1, 5, 251, 110251, 191448941, 904434761801, 9459612561834055, 191593734298902552191, 6835386432791154682927481, 400218584926232312004573701101, 36402864165071086859006490971345651, 4922828438813493756340086555005103394355
Offset: 0

Views

Author

Seiichi Manyama, May 19 2019

Keywords

Links

Crossrefs

Row n=4 of A308292.

Programs

  • Mathematica
    Table[Total[CoefficientList[Series[(1 + x + x^2/2 + x^3/6 + x^4/24)^n, {x, 0, 4*n}], x] * Range[0, 4*n]!], {n, 0, 15}] (* Vaclav Kotesovec, May 24 2020 *)
  • PARI
    {a(n) = sum(i=0, 4*n, i!*polcoef(sum(j=0, 4, x^j/j!)^n, i))}

Formula

a(2) = binomial(0+0,0) + binomial(0+1,1) + binomial(0+2,2) + binomial(0+3,3) + binomial(0+4,4) + binomial(1+0,0) + binomial(1+1,1) + binomial(1+2,2) + binomial(1+3,3) + binomial(1+4,4) + binomial(2+0,0) + binomial(2+1,1) + binomial(2+2,2) + binomial(2+3,3) + binomial(2+4,4) + binomial(3+0,0) + binomial(3+1,1) + binomial(3+2,2) + binomial(3+3,3) + binomial(3+4,4) + binomial(4+0,0) + binomial(4+1,1) + binomial(4+2,2) + binomial(4+3,3) + binomial(4+4,4) = 251.
a(n) ~ sqrt(Pi) * 2^(5*n + 3/2) * n^(4*n + 1/2) / (3^n * exp(4*n - 1)). - Vaclav Kotesovec, May 24 2020
Showing 1-5 of 5 results.

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