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.

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A351279 a(n) = Sum_{k=0..n} 2^k * k^(n-k).

Original entry on oeis.org

1, 2, 6, 18, 58, 202, 762, 3114, 13754, 65386, 332922, 1806506, 10398266, 63226858, 404640250, 2716838186, 19083233210, 139874994282, 1067462826874, 8464760754602, 69620304280890, 592925117961450, 5220996124450042, 47467755352580650, 445027186867923642
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2022

Keywords

Links

Crossrefs

Cf. A003101, A026898, A038125, A349962.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[2^k * k^(n-k), {k, 1, n}]; Array[a, 25, 0] (* Amiram Eldar, Feb 06 2022 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*k^(n-k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)))

Formula

G.f.: Sum_{k>=0} (2*x)^k/(1 - k*x).
a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/2))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/2) - n) / LambertW(exp(1)*n/2)^(n + 1/2). - Vaclav Kotesovec, Feb 06 2022

A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

Original entry on oeis.org

1, 2, 6, 45, 1051, 88602, 27121964, 37004504305, 198705527223757, 5595513387083114570, 686714367475480207331582, 468422339816915120237104999421, 1664212116512828935888786624225704855
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2023

Keywords

Links

Crossrefs

Cf. A026898, A349893, A359658.
Cf. A031971, A349836, A349883.
Cf. A003101, A349882.

Programs

  • PARI
    a(n) = sum(k=0, n, k^(k*(n-k+1)));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^k*x)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k+1)^(k+1)*x)))

Formula

G.f.: Sum_{k>=0} (k * x)^k/(1 - k^k * x).
G.f.: Sum_{k>=0} x^k/(1 - (k+1)^(k+1) * x).
a(n) = A349893(n+1) - 1.

A361476 Antidiagonal sums of A361475.

Original entry on oeis.org

0, 1, 4, 12, 34, 99, 308, 1040, 3820, 15197, 65060, 297828, 1449742, 7468527, 40555732, 231335944, 1381989864, 8623700793, 56078446596, 379233142780, 2662013133274, 19362917621979, 145719550012276, 1133023004941248, 9090156910550084, 75161929739797493, 639793220877941476
Offset: 0

Views

Author

Stefano Spezia, Mar 13 2023

Keywords

Links

Crossrefs

Cf. A026898, A104878, A361475.

Programs

  • Maple
    f:= proc(n) local k;
add( (k^(n-k+2) - 1)/(k - 1),k=2..n+2)
end proc:
map(f, [$0..30]); # Robert Israel, Nov 12 2024
  • Mathematica
    A361475[n_,k_]:=(k^n-1)/(k-1); a[n_]:=Sum[A361475[n-k+2,k],{k,2,n+2}]; Array[a,27,0]

Formula

a(n) = Sum_{k=2..n+2} (k^(n-k+2) - 1)/(k - 1).
a(n) ~ A026898(n).
a(n) = Sum_{k=0..n} k * A104878(n,k). - Alois P. Heinz, Dec 05 2023

A367011 a(n) = Sum_{k=0..n} k! * k^(n-k).

Original entry on oeis.org

1, 1, 3, 11, 51, 287, 1899, 14447, 124251, 1192127, 12623979, 146250287, 1840024251, 24983863967, 364140992139, 5670546353807, 93960923507931, 1650688221777407, 30646388716777899, 599565840087487727, 12328458398407260411
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2023

Keywords

Links

Crossrefs

Cf. A026898, A112541, A367012.
Cf. A003422, A233449, A368555.

Programs

  • Mathematica
    Table[Sum[k! * k^(n-k), {k, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=0, n, k!*k^(n-k)); \\ Seiichi Manyama, Dec 31 2023

Formula

a(n) ~ Pi * n^(n+1) / exp(n).
a(n) ~ sqrt(Pi*n/2) * n!.

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 31 2023

A367012 a(n) = Sum_{k=0..n} k! * (n-k)^k.

Original entry on oeis.org

1, 1, 2, 5, 18, 95, 704, 6945, 87254, 1349603, 25064700, 548782229, 13970248610, 408882114519, 13625250384488, 512421111644105, 21577659567580014, 1010231138742981515, 52263989531636074964, 2971798406660674944573, 184850941269122564302010
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2023

Keywords

Links

Crossrefs

Cf. A005193, A026898, A367011.

Programs

  • Mathematica
    Table[Sum[k! * (n-k)^k, {k, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=0, n, k!*(n-k)^k); \\ Seiichi Manyama, Dec 31 2023

Formula

log(a(n)) ~ n*(2*log(n) - log(log(n)) - 2 - log(2) + log(2*log(n))/(2*log(n)) + 1/(8*log(n)^2)).

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 31 2023

A156354 Triangle T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 08 2009

Keywords

Comments

This sequence is an approximation of Pascal's triangle with interior Kurtosis.
Essentially the same as A055652. - R. J. Mathar, Feb 19 2009

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,   8,    4,     1;
  1,  5,  17,   17,     5,     1;
  1,  6,  32,   54,    32,     6,     1;
  1,  7,  57,  145,   145,    57,     7,     1;
  1,  8, 100,  368,   512,   368,   100,     8,    1;
  1,  9, 177,  945,  1649,  1649,   945,   177,    9,   1;
  1, 10, 320, 2530,  5392,  6250,  5392,  2530,  320,  10,  1;
  1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1;
The interior Kurtosis, T(n,k) - binomial(n, k), is:
  0;
  0, 0;
  0, 0,   0;
  0, 0,   0,    0;
  0, 0,   2,    0,     0;
  0, 0,   7,    7,     0,     0;
  0, 0,  17,   34,    17,     0,     0;
  0, 0,  36,  110,   110,    36,     0,     0;
  0, 0,  72,  312,   442,   312,    72,     0,    0;
  0, 0, 141,  861,  1523,  1523,   861,   141,    0,   0;
  0, 0, 275, 2410,  5182,  5998,  5182,  2410,  275,   0, 0;
  0, 0, 538, 6908, 18455, 22939, 22939, 18455, 6908, 538, 0, 0;

Links

Crossrefs

Cf. A026898.

Programs

  • Magma
    [k eq 0 select 1 else k^(n-k) + (n-k)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2021
  • Mathematica
    T[n_, k_]:= If[n==0, 1, (k^(n-k) + (n-k)^k)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
  • Sage
    flatten([[1 if k==n else k^(n-k) + (n-k)^k for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 07 2021

Formula

T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1.
T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n,k) = [n=0] + 2*A026898(n-1). - G. C. Greubel, Mar 07 2021

Extensions

Edited by G. C. Greubel, Mar 07 2021

A245389 G.f. satisfies: A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)).

Original entry on oeis.org

1, 2, 6, 23, 102, 496, 2570, 13959, 78682, 457243, 2727360, 16647048, 103759186, 659500772, 4271197824, 28175622291, 189321228022, 1296246842443, 9049626101836, 64481397834665, 469461395956168, 3497006117588399, 26688813841105524, 208977790442594368, 1680981707733908594
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2014

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 102*x^4 + 496*x^5 + 2570*x^6 +...
where we have the following series identity:
A(x) = 1/(1-x*A(x)) + x/(1-2*x*A(x)) + x^2/(1-3*x*A(x)) + x^3/(1-4*x*A(x)) + x^4/(1-5*x*A(x)) + x^5/(1-6*x*A(x)) + x^6/(1-7*x*A(x)) +...
is equal to
A(x) = 1/(1-x) + x/(1-x)^2*A(x)/(1+x*A(x)) + 2!*x^2/(1-x)^3*A(x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3/(1-x)^4*A(x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4/(1-x)^5*A(x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5/(1-x)^6*A(x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...

Links

Crossrefs

Cf. A026898, A316367.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, x^m/(1-(m+1)*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, m!*x^m*A^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x*A +x*O(x^n))));polcoeff(, n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)).
(2) A(x) = Sum_{n>=0} n! * x^n/(1-x)^(n+1) * A(x)^n / Product_{k=1..n} (1 + k*x*A(x)).

A316367 G.f. satisfies: A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)^2).

Original entry on oeis.org

1, 2, 8, 45, 297, 2144, 16398, 130622, 1072509, 9015741, 77229624, 671868010, 5921241337, 52764270015, 474699837123, 4306641596007, 39363068782364, 362191362113221, 3352866324085927, 31210685632641522, 292025240058727496, 2745513045893833352, 25929804402647536603, 245958435266263341412, 2342884864036837008480, 22409497495190975013498
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 45*x^3 + 297*x^4 + 2144*x^5 + 16398*x^6 + 130622*x^7 + 1072509*x^8 + 9015741*x^9 + 77229624*x^10 + ...
where we have the following series identity:
A(x) = 1/(1-x*A(x)^2) + x/(1-2*x*A(x)^2) + x^2/(1-3*x*A(x)^2) + x^3/(1-4*x*A(x)^2) + x^4/(1-5*x*A(x)^2) + x^5/(1-6*x*A(x)^2) + x^6/(1-7*x*A(x)^2) + ...
is equal to
A(x) = 1/(1-x) + x/(1-x)^2*A(x)^2/(1+x*A(x)^2) + 2!*x^2/(1-x)^3*A(x)^4/((1+x*A(x)^2)*(1+2*x*A(x)^2)) + 3!*x^3/(1-x)^4*A(x)^6/((1+x*A(x)^2)*(1+2*x*A(x)^2)*(1+3*x*A(x)^2)) + 4!*x^4/(1-x)^5*A(x)^8/((1+x*A(x)^2)*(1+2*x*A(x)^2)*(1+3*x*A(x)^2)*(1+4*x*A(x)^2)) + 5!*x^5/(1-x)^6*A(x)^10/((1+x*A(x)^2)*(1+2*x*A(x)^2)*(1+3*x*A(x)^2)*(1+4*x*A(x)^2)*(1+5*x*A(x)^2)) + ...

Links

Crossrefs

Cf. A026898, A245389.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/(1 - (m+1)*x*A^2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(2*m)/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1 + k*x*A^2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)^2).
(2) A(x) = Sum_{n>=0} n! * x^n/(1-x)^(n+1) * A(x)^(2*n) / Product_{k=1..n} (1 + k*x*A(x)^2).

A322875 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals two.

Original entry on oeis.org

0, 1, 5, 21, 86, 361, 1584, 7315, 35635, 183080, 990659, 5635021, 33622161, 209973099, 1369560267, 9310957518, 65852852210, 483672626464, 3683088047043, 29033382412670, 236591717703447, 1990467019391404, 17268021545339042, 154304401318961489
Offset: 2

Views

Author

Alois P. Heinz, Dec 29 2018

Keywords

Links

Crossrefs

Column k=2 of A287215.
Cf. A026898, A287252.

Programs

  • Maple
    b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
a:= n-> (k-> A(n, k)-A(n, k-1))(2):
seq(a(n), n=2..30);
  • Mathematica
    b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
a[n_] := With[{k = 2}, A[n, k] - A[n, k - 1]];
a /@ Range[2, 30] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) = A287252(n) - A026898(n-1).

A351340 a(n) = Sum_{k=0..n} n^k * k^(n-k).

Original entry on oeis.org

1, 1, 6, 48, 516, 6955, 112686, 2132634, 46167560, 1125116901, 30481672610, 908760877244, 29565986232396, 1042354163621927, 39584173937284438, 1610922147768721590, 69940319175066857488, 3226793787576474492657, 157649292247463953189578
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2022

Keywords

Links

Crossrefs

Main diagonal of A351339.
Cf. A026898, A031973, A155956, A303991.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[n^k * k^(n - k), {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 08 2022 *)
  • PARI
    a(n) = sum(k=0, n, n^k*k^(n-k));

Formula

a(n) = [x^n] Sum_{k>=0} (n*x)^k/(1 - k*x).
a(n) ~ c * n^(n + 1/2), where c = sqrt(Pi)/2. - Vaclav Kotesovec, Feb 09 2022
Previous Showing 31-40 of 45 results. Next

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