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 10 results.

A066166 Stanley's children's game. Class of n (named) children forms into rings with exactly one child inside each ring. We allow the case when outer ring has only one child. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.

Original entry on oeis.org

2, 3, 20, 90, 594, 4200, 34544, 316008, 3207240, 35699400, 432690312, 5672581200, 79991160144, 1207367605080, 19423062612480, 331770360922560, 5997105160795584, 114373526841360000, 2295170834453089920
Offset: 2

Views

Author

Len Smiley, Dec 12 2001

Keywords

Comments

Apparently n divides a(n), so a(n)/n = 1, 1, 5, 18, 99, 600, 4318, 35112, 320724, 3245400, 36057526, 436352400, 5713654296, ... - R. J. Mathar, Oct 31 2015

Examples

			a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Crossrefs

Cf. A066165. Apart from initial terms and signs, same as A007113.
Cf. A343579.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-1+1/(1-x)^x)); [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Aug 29 2018
  • Mathematica
    Drop[With[{nn=20},CoefficientList[Series[1/(1-x)^x-1,{x,0,nn}],x] Range[ 0,nn]!],2] (* Harvey P. Dale, Sep 17 2011 *)
  • Maxima
    b(n):=if n=0 then 1 else (n-1)!*sum((1+1/i)*b(n-i-1)/(n-i-1)!,i,1,n-1);
makelist(a(n),n,2,10); /* Vladimir Kruchinin, Feb 25 2015 */
  • PARI
    a(n)=if(n<0,0,n!*polcoeff(-1+1/(1-x+x*O(x^n))^x,n))
  • PARI
    {a(n) = n!*polcoeff( sum(m=1,n, x^m/m! * prod(k=0,m-1,x + k) +x*O(x^n) ), n)}
for(n=2,20, print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2015
  • PARI
    a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 10 2022

Formula

E.g.f.: -1+1/(1-x)^x.
a(n) ~ n! * (1 - 1/n + (1-log(n)-gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014
a(n) = b(n), n>0, a(0)=0, where b(n) = (n-1)!*Sum_{i=1..n-1} (1+1/i)*b(n-i-1)/(n-i-1)!, b(0)=1. - Vladimir Kruchinin, Feb 25 2015
E.g.f.: Sum_{n>=1} x^n/n! * Product_{k=0..n-1} (k + x). - Paul D. Hanna, Oct 26 2015
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 10 2022

A353252 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 2 * x).

Original entry on oeis.org

1, 0, 2, 2, 8, 24, 100, 488, 2832, 19096, 147296, 1281392, 12422864, 132870368, 1554525152, 19750621216, 270817685568, 3986140113792, 62686410981696, 1048946532137216, 18608550117641728, 348854564104019072, 6891109834644748032, 143058034748452036352
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2022

Keywords

Links

Crossrefs

Cf. A352802, A343579, A353253, A353254.
Cf. A097341.

Programs

  • Mathematica
    a[n_] := Sum[2^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+2*x)))
  • PARI
    a(n) = sum(k=0, n\2, 2^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} 2^k * |Stirling1(n-k,k)|.

A353253 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - x).

Original entry on oeis.org

1, 0, -1, -1, -1, -3, -14, -76, -480, -3491, -28792, -265708, -2713753, -30395515, -370509784, -4883351213, -69205187838, -1049436525897, -16956113955333, -290817728309779, -5277059794403117, -101005287980087110, -2033813167589257170, -42977173319758429942
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2022

Keywords

Crossrefs

Cf. A343579, A353252, A353254.
Cf. A353260.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j-x)))
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * |Stirling1(n-k,k)|.

A353254 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - 2 * x).

Original entry on oeis.org

1, 0, -2, -2, 0, 0, -12, -88, -608, -4664, -40032, -381200, -3993520, -45685472, -566975456, -7589393568, -109019255360, -1673050977024, -27321358963904, -473094230383616, -8659054324278528, -167044915214322816, -3387793305708038400, -72061754672510128384
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2022

Keywords

Crossrefs

Cf. A343579, A353252, A353253.
Cf. A353261.

Programs

  • Mathematica
    a[n_] := Sum[(-2)^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j-2*x)))
  • PARI
    a(n) = sum(k=0, n\2, (-2)^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * |Stirling1(n-k,k)|.

A357901 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n - 2*k,k)|.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 7, 27, 131, 771, 5320, 42119, 376174, 3740018, 40956593, 489749100, 6348744124, 88677555115, 1327628770657, 21208195526882, 360053293342379, 6473501562355779, 122874692176838047, 2455382300127368557, 51524333987938459606, 1132787775301639812263
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2022

Keywords

Crossrefs

Cf. A000142, A343579, A357902.

Programs

  • PARI
    a(n) = sum(k=0, n\3, abs(stirling(n-2*k, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x^2)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + x^2).

A357902 a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n - 3*k,k)|.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 6, 25, 123, 731, 5090, 40595, 364650, 3641903, 40026609, 480029801, 6237662582, 87298953249, 1309161984315, 20942605407386, 355971044728635, 6406714801013007, 121715861296354116, 2434125806029297550, 51113325326999860554, 1124432395936987325868
Offset: 0

Views

Author

Seiichi Manyama, Oct 19 2022

Keywords

Crossrefs

Cf. A000142, A343579, A357901.

Programs

  • PARI
    a(n) = sum(k=0, n\4, abs(stirling(n-3*k, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x^3)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + x^3).

A352802 Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 3 * x).

Original entry on oeis.org

1, 0, 3, 3, 15, 45, 198, 972, 5652, 37881, 289548, 2492640, 23906475, 253012653, 2930556024, 36883817127, 501315357690, 7318715960511, 114224260779891, 1897913866979529, 33449523840512127, 623265596538965334, 12241892922194658510, 252793167644378784006
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2022

Keywords

Links

Crossrefs

Cf. A343579, A353252, A353253, A353254.
Cf. A097342.

Programs

  • Mathematica
    a[n_] := Sum[3^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+3*x)))
  • PARI
    a(n) = sum(k=0, n\2, 3^k*abs(stirling(n-k, k, 1)));

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * |Stirling1(n-k,k)|.

A353289 a(n) = Sum_{k=0..floor(n/2)} (n-k)^k * |Stirling1(n-k,k)|.

Original entry on oeis.org

1, 0, 1, 2, 10, 51, 323, 2354, 19535, 181606, 1869549, 21110063, 259400501, 3445913273, 49207968328, 751698726580, 12231484211240, 211208935989003, 3857425360784596, 74292198980174828, 1504832580013205275, 31980327844846620785, 711498612995378484414
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2022

Keywords

Crossrefs

Cf. A343579, A352802, A353252.
Cf. A132393, A353290.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+k*x)))
  • PARI
    a(n) = sum(k=0, n\2, (n-k)^k*abs(stirling(n-k, k, 1)));

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + k * x).

A353290 a(n) = Sum_{k=0..floor(n/2)} (n-k)^(n-2*k) * |Stirling1(n-k,k)|.

Original entry on oeis.org

1, 0, 1, 2, 19, 393, 15177, 939394, 85063260, 10599342278, 1739073390797, 363404567436467, 94224446795779884, 29683590039199285223, 11167286542016941966714, 4945143125245884296040780, 2546112368234517215955646341, 1508197687055444623135714912377
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2022

Keywords

Crossrefs

Cf. A343579, A353255, A353256.
Cf. A353289.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, k*j+x)))
  • PARI
    a(n) = sum(k=0, n\2, (n-k)^(n-2*k)*abs(stirling(n-k, k, 1)));

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (k * j + x).

A357922 a(n) = Sum_{k=0..floor(n/5)} |Stirling1(n - 4*k,k)|.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 24, 121, 723, 5051, 40370, 363154, 3630565, 39929874, 479111219, 6228047601, 87188921464, 1307794924973, 20924276449014, 355707232027825, 6402657184129671, 121649439722758345, 2432972744390660437, 51092165603897459951
Offset: 0

Views

Author

Seiichi Manyama, Oct 20 2022

Keywords

Crossrefs

Cf. A000142, A343579, A357901, A357902.
Cf. A357920.

Programs

  • PARI
    a(n) = sum(k=0, n\5, abs(stirling(n-4*k, k, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x^4)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (j + x^4).
Showing 1-10 of 10 results.

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