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 1-10 of 11 results. Next

A033935 Sum of squares of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 6, 93, 2716, 127905, 8848236, 844691407, 106391894904, 17091486402849, 3410496772665940, 827540233598615691, 239946160014513220896, 81932406267721802925925, 32541656017173091541743368, 14874686717916861528415671285, 7753005946480818323895940923376
Offset: 0

Views

Author

Warren D. Smith, Dec 11 1999

Keywords

Comments

Two samples of size n are taken from an urn containing infinitely many marbles of n distinct colors. a(n)/n^(2*n) is the probability that the two samples match. That is, they contain the same number of each color of marbles without regard to order. - Geoffrey Critzer, Apr 19 2014

Links

Crossrefs

Column k=2 of A245397.
Main diagonal of A287316.
Cf. A364116.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A033935:= proc(n) series(hypergeom([],[1],z)^n, z=0, n+1): n!^2*coeff(%,z,n) end: seq(A033935(n), n=0..16); # Peter Luschny, May 31 2017
  • Mathematica
    Table[nn=n;n!^2 Coefficient[Series[(Sum[x^k/k!^2,{k,0,nn}])^n,{x,0,nn}],x^n],{n,1,20}] (* Geoffrey Critzer, Apr 19 2014 *)
Flatten[{1,Table[n!^2*Coefficient[Series[BesselI[0,2*Sqrt[x]]^n,{x,0,n}],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Jul 29 2014 *)
Table[SeriesCoefficient[HypergeometricPFQ[{},{1},x]^n, {x,0,n}] n!^2, {n,0,16}] (* Peter Luschny, May 31 2017 *)

Formula

a(n) is coefficient of x^n in expansion of n!^2*(1 + x/1!^2 + x^2/2!^2 + x^3/3!^2 + ... + x^n/n!^2)^n. - Vladeta Jovovic, Jun 09 2000
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.1024237701057210364324371415246345951600138303179762223318873762632384990..., c = 0.487465475752598098146353111500372156824276600165331887960705498284416... - Vaclav Kotesovec, Jul 29 2014, updated Jul 10 2023
a(n) = n!^2 * [z^n] hypergeom([], [1], z)^n. - Peter Luschny, May 31 2017

Extensions

More terms from James Sellers, Jun 01 2000 and Vladeta Jovovic, Jun 05 2000
a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 10, 1, 0, 1, 4, 27, 56, 1, 0, 1, 5, 52, 381, 346, 1, 0, 1, 6, 85, 1192, 6219, 2252, 1, 0, 1, 7, 126, 2705, 36628, 111753, 15184, 1, 0, 1, 8, 175, 5136, 124405, 1297504, 2151549, 104960, 1, 0
Offset: 0

Views

Author

Peter Luschny, May 30 2017

Keywords

Comments

Let A_m(n,k) = (n!)^m [x^n] hypergeom([], [1,…,1], z)^k where [1,…,1] lists (m-1) times 1. These arrays can be seen as generalizations of the power functions n^k. For m = 1 -> A003992, m = 2 -> A287316, m = 3 -> A287698.
A_m(n,n) is the sum of m-th powers of coefficients in the full expansion of (z_1+z_2+...+z_n)^n (compare A245397).
A287696 provide polynomials and A287697 rational functions generating the columns of the array.

Examples

			Array starts:
k\n| 0  1    2       3       4         5           6             7
---|-------------------------------------------------------------------
k=0| 1, 0,   0,      0,      0,        0,          0,            0, ... A000007
k=1| 1, 1,   1,      1,      1,        1,          1,            1, ... A000012
k=2| 1, 2,  10,     56,    346,     2252,      15184,       104960, ... A000172
k=3| 1, 3,  27,    381,   6219,   111753,    2151549,     43497891, ... A141057
k=4| 1, 4,  52,   1192,  36628,  1297504,   50419096,   2099649808, ... A287699
k=5| 1, 5,  85,   2705, 124405,  7120505,  464011825,  33031599725, ...
k=6| 1, 6, 126,   5136, 316206, 25461756, 2443835736, 263581282656, ...
       A001107,A287702,A287700,  A287701,                               A055733

Crossrefs

Rows: A000007 (k=0), A000012 (k=1), A000172 (k=2), A141057 (k=3), A287699 (k=4).
Columns: A000172 (n=1), A001477(n=1), A001107 (n=2), A287702 (n=3), A287700 (n=4), A287701 (n=5).
Cf. A055733 (diagonal), A287696, A287697, A003992, A287316, A245397.

Programs

  • Maple
    A287698_row := (k, len) -> seq(A287696_poly(j)(k), j=0..len):
A287698_row := proc(k, len) hypergeom([], [1, 1], x):
series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
for k from 0 to 6 do A287698_row(k, 9) od;
A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287698_col(n, 9) od;
  • Mathematica
    Table[Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)

A055733 Sum of third powers of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 10, 381, 36628, 7120505, 2443835736, 1351396969615, 1127288317316008, 1349611750487720817, 2230372438317527996620, 4930842713588476723120511, 14211567663513739084746570600, 52259895270824126097423028107277, 240736564755509319272061470644316416
Offset: 0

Views

Author

Vladeta Jovovic, Jun 09 2000

Keywords

Links

Crossrefs

Cf. A033935.
Column k=3 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^2/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A055733 := proc(n) series(hypergeom([],[1,1],z)^n,z=0,n+1): n!^3*coeff(%,z,n) end: seq(A055733(n), n=0..14); # Peter Luschny, May 31 2017
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^2 / j!, {j, 0, n}]]]; a[n_] := n!*b[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)
Table[SeriesCoefficient[HypergeometricPFQ[{}, {1,1}, x]^n, {x,0,n}] n!^3, {n,0,14}] (* Peter Luschny, May 31 2017 *)

Formula

a(n) is coefficient of x^n in expansion of n!^3*(1+x/1!^3+x^2/2!^3+x^3/3!^3+...+x^n/n!^3)^n.
a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 1.74218173246413..., c = 0.5728782413434... . - Vaclav Kotesovec, Aug 20 2014
a(n) = (n!)^3 * [z^n] hypergeom([], [1, 1], z)^n. - Peter Luschny, May 31 2017

Extensions

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A245398 Sum of n-th powers of coefficients in full expansion of (z_1 + z_2 + ... + z_n)^n.

Original entry on oeis.org

1, 1, 6, 381, 591460, 41262262505, 207874071367118436, 110807909819808911886548575, 8558639841332633529404511878004186120, 124773193097402414339622625011223384066643153613969, 431220070110830123225191271755402469908417673177630594034899052340
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2014

Keywords

Links

Crossrefs

Main diagonal of A245397.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..12);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
   Sum[b[n-j, i-1, k]*Binomial[n, j]^(k-1)/j!, {j, 0, n}]]];
a[n_] := n!*b[n, n, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jun 27 2022, after Alois P. Heinz *)

Formula

a(n) = [x^n] (n!)^n * (Sum_{j=0..n} x^j/(j!)^n)^n.
a(n) = A245397(n,n).

A055740 Sum of fourth powers of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 18, 1785, 591460, 495872505, 882463317636, 2956241639184631, 17088644286346128840, 159584255348964655673745, 2286523844910576580400966980, 48220116744252542032928364578451, 1446485887751234540636003724054342864, 59981372975740557234356339667492583487125
Offset: 0

Views

Author

Vladeta Jovovic, Jun 09 2000

Keywords

Links

Crossrefs

Cf. A033935.
Column k=4 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^3/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^3 /j!, {j, 0, n}]]]; a[n_] := n!*b[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

a(n) is coefficient of x^n in expansion of n!^4*(1+x/1!^4+x^2/2!^4+x^3/3!^4+...+x^n/n!^4)^n.
a(n) ~ c * d^n * (n!)^4 / sqrt(n), where d = 1.511958716403..., c = 0.6632048858... . - Vaclav Kotesovec, Aug 20 2014

Extensions

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A246240 Sum of fifth powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n.

Original entry on oeis.org

1, 1, 34, 9237, 11007556, 41262262505, 393602334214536, 8250608306349317503, 341379009411431516029576, 25693424488177173143564108049, 3298778490446719483156753593432700, 686045693667123232536420797701863401231, 221475400673152122602874526565943771742514376
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2014

Keywords

Links

Crossrefs

Column k=5 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^4/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..15);

Formula

a(n) = [x^n] (n!)^5 * (Sum_{j=0..n} x^j/(j!)^5)^n.

A246241 Sum of sixth powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n.

Original entry on oeis.org

1, 1, 66, 51033, 227263876, 3942914312505, 207874071367118436, 28034487027123336138967, 8522964991458712709499563784, 5302659152501095787067079018931409, 6255441983177258421672575234559926069140, 13154762734940720943667470423246456789300752691
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2014

Keywords

Links

Crossrefs

Column k=6 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^5/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..15);

Formula

a(n) = [x^n] (n!)^6 * (Sum_{j=0..n} x^j/(j!)^6)^n.

A246242 Sum of seventh powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n.

Original entry on oeis.org

1, 1, 130, 293061, 5018329348, 414999981562505, 124389170238814179336, 110807909819808911886548575, 253626563859350391170222920686088, 1334380576777390115212093953972864348177, 14777734823564325121187478504310896072495827020
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2014

Keywords

Links

Crossrefs

Column k=7 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^6/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..15);

Formula

a(n) = [x^n] (n!)^7 * (Sum_{j=0..n} x^j/(j!)^7)^n.

A246243 Sum of eighth powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n.

Original entry on oeis.org

1, 1, 258, 1718985, 115245958660, 46377854607812505, 80785609177262537107236, 486005483266096999009285275991, 8558639841332633529404511878004186120, 388791577542234912413815089860741309780872785, 41231194444310047390596429351583294775856761836687780
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2014

Keywords

Links

Crossrefs

Column k=8 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^7/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..15);

Formula

a(n) = [x^n] (n!)^8 * (Sum_{j=0..n} x^j/(j!)^8)^n.

A246244 Sum of ninth powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n.

Original entry on oeis.org

1, 1, 514, 10195797, 2703788516356, 5361940142039062505, 55063667396158825603112136, 2272169230481993564658922073502463, 312894608313254360747865383525129561090056, 124773193097402414339622625011223384066643153613969
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2014

Keywords

Links

Crossrefs

Column k=9 of A245397.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^9, j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);

Formula

a(n) = [x^n] (n!)^9 * (Sum_{j=0..n} x^j/(j!)^9)^n.
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