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 1-9 of 9 results.

A132608 Self-convolution square-root of A062817 (offset 2); thus g.f. A(x) satisfies: A(x)^2 = Sum_{n>=2} A062817(n)*x^n, where A062817(n) = Sum_{k=0..n} (n-k)^k*k^(n-k).

Original entry on oeis.org

1, 2, 9, 58, 469, 4530, 50491, 634790, 8861043, 135750454, 2262315973, 40726646802, 787471241647, 16275700505510, 358103286781293, 8357593147404346, 206241859929682177, 5366082228239257410
Offset: 1

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Examples

			A(x) = x + 2x^2 + 9x^3 + 58x^4 + 469x^5 + 4530x^6 +...+ a(n)*x^n +...
A(x)^2 = x^2 + 4x^3 + 22x^4 + 152x^5 + 1251x^6 +...+ A062817(n)*x^n +...

Links

Crossrefs

Cf. A062817, A132609.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(Sum[x^(k-2) * Sum[(k-j)^j * j^(k-j), {j, 0, k}], {k, 1, 2*nmax}])^(1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 22 2021 *)
  • PARI
    {a(n)=polcoeff((sum(m=2,n+1,sum(k=0,m,(m-k)^k*k^(m-k))*x^m +x*O(x^(n+1))))^(1/2),n)}

Formula

a(n) ~ exp(1) * sqrt(2*Pi/3) * n^(n + 3/2) / 2^(n+3). - Vaclav Kotesovec, Nov 22 2021

A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

Original entry on oeis.org

1, 0, 2, 12, 120, 1480, 22320, 396564, 8118656, 188185680, 4871980800, 139342178140, 4363291266048, 148470651659928, 5455056815237120, 215238256785814500, 9077047768435752960, 407449611073696325536, 19396232794530856894464, 976025303642559490903980
Offset: 0

Views

Author

Paul D. Hanna, Oct 11 2007

Keywords

Comments

E.g.f. equals the square of the e.g.f. of A060435, where A060435(n) = number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

Examples

			E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).

Links

Crossrefs

Cf. A060435; indirectly related: A062817, A132608.
Cf. A063170, A277458, A277490, A277510.

Programs

  • Maple
    seq(simplify(GAMMA(n+1,-n)*(-exp(-1))^n),n=0..20); # Vladeta Jovovic, Oct 17 2007
  • Mathematica
    CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
Table[a[n], {n, 0, 20}] (* Gerry Martens, May 05 2016 *)
  • PARI
    {a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))}
  • PARI
    /* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
{a(n)=my(LambertW=-x*sum(k=0,n,(-x)^k*(k+1)^(k-1)/k!) +x*O(x^n)); n!*polcoeff(1/(1-subst(LambertW,x,-x)^2),n)}

Formula

a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).
a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*n^k/k!. - Vladeta Jovovic, Oct 17 2007
a(n) ~ n^n/2. - Vaclav Kotesovec, Nov 27 2012, simplified Nov 22 2021
a(n) = n! * [x^n] exp(n*x)/(1 + x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - Thomas Scheuerle, Jan 29 2024

A100262 Expansion of A(x)^2, where A(x) = o.g.f. of n^n (A000312).

Original entry on oeis.org

1, 2, 9, 62, 582, 6978, 102339, 1779222, 35809052, 819103178, 20987183525, 595341928814, 18519658804818, 626784970780690, 22926284614808071, 901188628763393606, 37882728189752349304, 1695744102631158083866
Offset: 0

Views

Author

Ralf Stephan, Nov 20 2004

Keywords

Examples

			(1 + x + 4x^2 + 27x^3 + 256x^4 +...)^2 = 1 + 2x + 9x^2 + 62x^3 +...

Links

Crossrefs

Cf. A000312, A053729, A062817, A349874.

Programs

  • Mathematica
    nn=17;f[x_]=1+Sum[n^n x^n,{n,1,nn}];CoefficientList[Series[f[x]^2,{x,0,nn}],x] (* Geoffrey Critzer, Nov 05 2013 *)
  • PARI
    a(n) = sum(k=0, n, k^k*(n-k)^(n-k)); \\ Seiichi Manyama, Dec 03 2021

Formula

a(n) = Sum_{k=0..n} k^k * (n-k)^(n-k). - Tilman Neumann, Dec 13 2008
a(n) ~ 2 * n^n. - Vaclav Kotesovec, Dec 03 2021

A132609 Antidiagonal sum of table A072590(n,k) = n^(k-1)*k^(n-1) for n>=1.

Original entry on oeis.org

1, 2, 6, 26, 147, 1026, 8532, 82394, 906485, 11194402, 153347766, 2307805402, 37851581159, 672037936898, 12841521329896, 262772642843802, 5733086299727913, 132853067341477538, 3258726189638877610
Offset: 1

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Comments

A072590(n,k) equals the number of spanning trees in complete bipartite graph K(n,k).
Also the number of minimum connected dominating sets of the (n+1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Jun 03 2024 and Mar 05 2025

Links

Crossrefs

Cf. A072590, A062817; A132608.

Programs

  • Mathematica
    Table[Sum[(n - k + 1)^(k - 1) k^(n - k), {k, n}], {n, 30}] (* Harvey P. Dale, Jun 26 2021 *)
  • PARI
    a(n)=sum(k=1,n,(n-k+1)^(k-1)*k^(n-k))

Formula

a(n) = Sum_{k=1..n} (n-k+1)^(k-1)*k^(n-k) for n>=1.
a(n) ~ sqrt(2*Pi/3) * exp(1) * n^(n - 1/2) / 2^n. - Vaclav Kotesovec, Nov 22 2021

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625
Offset: 0

Views

Author

Henry Bottomley, Jul 02 2001

Keywords

Comments

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

Examples

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Links

Crossrefs

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1
Cf. A055651, A055652, A303990.

Programs

  • Mathematica
    {{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)
  • PARI
    t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

Formula

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A252708 Sum_{k=0..n} k^(n+k)*(n+k)^k.

Original entry on oeis.org

2, 259, 158268, 269187906, 978285792294, 6507711545263297, 71556972889550923184, 1209737575506373416337924, 29788382281118324885770212522, 1024436101844282730312688949278959, 47581802360081733847924522879325619588, 2904105920298407789174888476843692390201446
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 20 2014

Keywords

Crossrefs

Cf. A062817, A252709, A252710.

Programs

  • Mathematica
    Table[Sum[k^(n+k)*(n+k)^k, {k, 0, n}], {n, 1, 12}]

Formula

a(n) ~ 2^n * n^(3*n). - Vaclav Kotesovec, Dec 20 2014

A252709 Sum_{k=0..n} k^(n+k)*(n-k)^k.

Original entry on oeis.org

0, 1, 34, 2446, 315788, 66140883, 21216724582, 10063816019004, 6854671838281696, 6482951652242448021, 8197210060548767881834, 13365577359964731097098090, 27292891758122129732383863668, 68341049055686519197950975142247, 206830351842829303737616801603098478
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 20 2014

Keywords

Crossrefs

Cf. A062817, A252708, A252710.

Programs

  • Mathematica
    Table[Sum[k^(n+k)*(n-k)^k, {k, 0, n}], {n, 1, 15}]

Formula

log(a(n)) ~ n*(3*log(n)- log(log(n)) - 1 - log(2) + (log(log(n)) + log(2) - 2) / (2*log(n))). - Vaclav Kotesovec, Nov 22 2021

A252710 Sum_{k=0..n} k^(n-k)*(n+k)^k.

Original entry on oeis.org

2, 19, 270, 5274, 131250, 3971953, 141615782, 5814096100, 270145723914, 14014491434175, 802959561033630, 50357860270776302, 3431283333644097698, 252413829036827831821, 19937800344936448113750, 1683065517269356710047112, 151213514611062314791034874
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 20 2014

Keywords

Crossrefs

Cf. A062817, A252708, A252709.

Programs

  • Mathematica
    Table[Sum[k^(n-k)*(n+k)^k, {k, 0, n}], {n, 1, 20}]

Formula

a(n) ~ 2^n * n^n / (1-exp(-1/2)/2). - Vaclav Kotesovec, Dec 20 2014

A351768 a(n) = n! * Sum_{k=0..n} k^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 0, 2, 18, 276, 6260, 190950, 7523082, 371286440, 22356290952, 1608686057610, 136069954606190, 13345029902628732, 1500054487474871484, 191349476316804534638, 27464505325501082617170, 4402551348139824475260240, 783025812197886669354545552
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Cf. A006153, A062817, A134095, A351765, A351780.

Programs

  • Mathematica
    Join[{1}, Table[n!*Sum[k^(n-k) * (n-k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*(n-k)^k/k!);

Formula

log(a(n)) ~ n *(2*log(n) - log(log(n)) - 2 + (log(log(n)) + log(log(n)-1) + 1)/log(n)). - Vaclav Kotesovec, Feb 19 2022
Showing 1-9 of 9 results.

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