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.

Previous Showing 11-19 of 19 results.

A382739 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n,k)^2.

Original entry on oeis.org

1, 4, 44, 1084, 48044, 3281404, 316032044, 40592233084, 6687195379244, 1372291071723004, 342877475325619244, 102409872018962876284, 36014541870868393113644, 14724003012156426011095804, 6922777830859189006847193644, 3708347961746448904830944962684
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Crossrefs

Main diagonal of A382736.
Cf. A048144, A382737, A382738.
Cf. A382678.

Programs

  • PARI
    a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n, k, 2)^2);

Formula

a(n) == 0 (mod 4) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^4.

A182553 Chromatic invariant of the complete tripartite graph K_(n,n,n).

Original entry on oeis.org

1, 11, 1243, 490043, 463370491, 860454250571, 2769263554592683, 14178247400433059003, 108483732651999512059291, 1182804548772797481324575531, 17700419121823142496192223238923, 352709466470858225716888461028622363, 9127611521817307582541815420363992765691
Offset: 1

Views

Author

Alois P. Heinz, May 04 2012

Keywords

Comments

The chromatic invariant equals the absolute value of the first derivative of the chromatic polynomial evaluated at 1.

Links

Crossrefs

Cf. A008277, A048144.

Programs

  • Maple
    P:= n-> expand(add(add(Stirling2(n, k) *Stirling2(n, m)
        *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)):
a:= n-> abs(subs(q=1, diff(P(n), q))):
seq(a(n), n=1..15);
  • Mathematica
    Table[Sum[StirlingS2[n, k] StirlingS2[n, m] (-1)^(k + m + n) (1 - k - m)^n Gamma[k + m - 1], {k, n}, {m, n}], {n, 10}] (* Eric W. Weisstein, Apr 26 2017 *)
  • PARI
    a(n)={my(s=vector(n, k, stirling(n,k,2))); sum(i=1,n, sum(j=1,n, sum(k=1,n, (-1)^(n+i+j+k)*s[i]*s[j]*s[k]*(i+j+k-2)! )))} \\ Andrew Howroyd, Apr 22 2018
  • PARI
    a(n)={(-1)^n*subst(serlaplace(sum(k=1,n,stirling(n,k,2)*x^k)^3/x^2),x,-1)} \\ Andrew Howroyd, Apr 22 2018

Formula

a(n) = |(d/dq P(n,q)){q=1}| with P(n,q) = Sum{k,m=1..n} S2(n,k) * S2(n,m) * (q-k-m)^n * Product_{i=0..k+m-1} (q-i) and S2 = A008277.
a(n) ~ (n-1)!^3 / (Pi * 3^(3/2) * (1 - log(3/2)) * (log(3/2))^(3*n-1)). - Vaclav Kotesovec, Sep 03 2014, updated Feb 18 2017
a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (-1)^(n+i+j+k) * Stirling2(n,i) * Stirling2(n,j) * Stirling2(n,k) * (i+j+k-2)!. - Andrew Howroyd, Apr 22 2018

A192564 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*Stirling2(n,k)*(k!)^2.

Original entry on oeis.org

1, 1, 5, 74, 2186, 106524, 7703896, 773034912, 102673179360, 17429291711280, 3680338415133024, 945958227345434016, 290761516548473591232, 105309706114422166775040, 44384982810939832477305600, 21536846291826596564956445184
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Crossrefs

Cf. A047793, A048144, A382792.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n,k]]StirlingS2[n,k]k!^2,{k,0,n}],{n,0,100}]
nmax = 20; Table[SeriesCoefficient[1/(1 + (E^x - 1)*Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, nmax}] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Apr 08 2025 *)
  • Maxima
    makelist(sum(abs(stirling1(n,k))*stirling2(n,k)*k!^2,k,0,n),n,0,24);

Formula

a(n) ~ c * LambertW(-1, -r*exp(-r))^n * n!^2 / (sqrt(n) * LambertW(-exp(-1/r)/r)^n), where r = 0.673313285145753168... is the root of the equation (1 + 1/(r*LambertW(-exp(-1/r)/r))) * (r + LambertW(-1, -r*exp(-r))) = 1 and c = 0.27034346270211507329954765593360596752557904498770241464597402478625037569... - Vaclav Kotesovec, Jul 05 2021
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + (exp(x) - 1) * log(1 - y)). - Ilya Gutkovskiy, Apr 06 2025

A375780 a(n) = Sum_{k=0..n} binomial(n,k) * (k! * S(n,k))^2, where S(,) are Stirling numbers of second kind.

Original entry on oeis.org

1, 1, 6, 147, 6940, 536405, 62352066, 10136833063, 2195583006072, 611230451090409, 212649006828729790, 90405046457569649531, 46115367523234055367828, 27797472578675758999950013, 19546873204979999617317371898, 15858780455222184878234284613775, 14703883436182303949571115531615216, 15450188317599029331216704733732600017
Offset: 0

Views

Author

Max Alekseyev, Aug 27 2024

Keywords

Links

Crossrefs

Cf. A048144.

Programs

  • Mathematica
    A375780[n_] := Sum[Binomial[n, k]*(k!*StirlingS2[n, k])^2, {k, 0, n}];
Array[A375780, 20, 0] (* Paolo Xausa, Nov 07 2024 *)
  • PARI
    { a375780(n) = sum(k=0,n, binomial(n,k) * (k!*stirling(n,k,2))^2); }

Formula

a(n) = n! * Sum_{k=0..n} k^n/k! * Sum_{m=0}^{n-k} (m+k)!/m!/(n-k-m)! * (-1)^m * S(n,m+k).
G.f.: the diagonal of 1 - t(x,y)*W'(t(x,y)), where t(x,y) := x*(1-e^y)*e^(x*(2-e^y)) and W() is Lambert W function.
a(n) ~ c * d^n * n^(2*n), where d = (2*r-1)^2*r/(exp(2)*(-1 + r + sqrt((1-r)*r))^2) = 0.522647981756854997298666108603651720918622906877425888529..., r = 0.665183670620587020892773716469052817866519211832581651... is the root of the equation (1-r)*(1 + r*LambertW(-1/(exp(1/r)*r)))^2 = r^3*LambertW(-1/(exp(1/r)*r))^2 and c = 1.38671243965876142096898080117513697606381035589463940412659515589... - Vaclav Kotesovec, Nov 07 2024

A382793 a(n) = Sum_{k=0..n} (-1)^k * (Stirling2(n,k) * k!)^2.

Original entry on oeis.org

1, -1, 3, -1, -525, 21599, -575757, -11712961, 4147828275, -478419026401, 27474795508083, 3849481231073279, -1772585499434165325, 366912253456842693599, -26525609280231515934477, -17189616925094873258825281, 10414911263566240831226298675, -3136992122810471155294591778401
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A047797, A048144, A192552.

Programs

  • Mathematica
    Table[Sum[(-1)^k (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(2 - Exp[x] - Exp[y] + Exp[x + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (2 - exp(x) - exp(y) + exp(x + y)).

A382794 a(n) = Sum_{k=0..n} Stirling1(n,k) * Stirling2(n,k) * (k!)^2.

Original entry on oeis.org

1, 1, 3, 2, -418, -14676, -234344, 18565056, 2659703616, 169046742960, -6539356064736, -4061128974843744, -672969012637199040, -19289566159655581440, 27323548725052131528960, 10157639436460221570630144, 1433264952547826545065237504, -520046813680980959472490690560
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A000670, A006252, A047792, A048144, A064618, A192554, A192564, A382792.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n, k] StirlingS2[n, k] (k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 - (Exp[x] - 1) Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - (exp(x) - 1) * log(1 + y)).

A291774 Triangle read by rows: chromatic invariant T(n,m) of the complete bipartite graph K_{m,n}.

Original entry on oeis.org

1, 0, 1, 0, 1, 5, 0, 1, 13, 73, 0, 1, 29, 301, 2069, 0, 1, 61, 1081, 11581, 95401, 0, 1, 125, 3613, 57749, 673261, 6487445, 0, 1, 253, 11593, 268381, 4306681, 55213453, 610093513, 0, 1, 509, 36301, 1191989, 25794781, 431525429, 6077248381, 75796724309, 0, 1, 1021, 111961, 5136061, 147587401, 3173843821, 56153444761, 864806272861, 12020754177001
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2017

Keywords

Examples

			Triangle begins:
1
0 1
0 1 5
0 1 13 73
0 1 29 301 2069

Links

Crossrefs

Main diagonal gives A048144.

Programs

  • Mathematica
    Join[{1}, Table[Sum[k! (-1)^(k + m) (k + 1)^n StirlingS2[m, k + 2], {k, 0, m - 1}], {n, 2, 10}, {m, n}]] // Flatten

Formula

T(m,n) = Sum_{k = 0..m-1} k!*(-1)^(k + m)*(k + 1)^n*Stirling2(m, k + 2) for max(m,n) > 1.

A382805 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling1(n,k) * k!)^2.

Original entry on oeis.org

1, 1, 3, 4, -272, -8524, -96596, 9634752, 983055168, 36429411456, -4303305703296, -1051644384152064, -89651253435644160, 10632887072757561600, 5599203549778990667520, 914684633796830925275136, -89559567563652079025946624, -104514775371103880549281775616
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A006252, A007840, A047796, A048144, A192554, A320502, A382792, A382793.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) (StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 + Log[1 + x] Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + log(1 + x) * log(1 - y)).

A382847 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (Stirling2(n,k) * k!)^2.

Original entry on oeis.org

1, 1, 14, 579, 48044, 6647405, 1379024730, 400315753159, 154879704709784, 77018569697097009, 47863427797633958630, 36348262891572161261963, 33119479438137288670256964, 35660343372397246917403353013, 44791475616825872944740798413234, 64911462519379469821754507087299215
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Crossrefs

Cf. A048144, A305919, A382737, A382738, A382739, A382853.

Programs

  • Mathematica
    Table[Sum[Binomial[n + k - 1, k] (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 15}]
Table[(n!)^2 SeriesCoefficient[1/(Exp[x] + Exp[y] - Exp[x + y])^n, {x, 0, n}, {y, 0, n}], {n, 0, 15}]
  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 2))^2); \\ Seiichi Manyama, Apr 06 2025

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y))^n.
a(n) == 0 (mod n) for n > 0. - Seiichi Manyama, Apr 06 2025
a(n) ~ c * (r*(1+r)*(1 + 2*r + 2*sqrt(r*(1+r))))^n * n^(2*n) / exp(2*n), where r = 0.78386040488712123296193324113250946749673854534386788724235... is the root of the equation r = (1+r) * (1 + 1/(r*LambertW(-exp(-1/r)/r)))^2 and c = 0.947509273452712778524331973956110163137127694168427319... - Vaclav Kotesovec, Apr 08 2025
Previous Showing 11-19 of 19 results.

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