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 1-5 of 5 results.

A357119 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 12, 120, 0, 1, 0, 0, 0, 6, 60, 720, 0, 1, 0, 0, 0, 1, 35, 360, 5040, 0, 1, 0, 0, 0, 0, 10, 226, 2520, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1645, 20160, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 13454, 181440, 3628800, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,  1,  1, 1, ...
  0,   1,   0,   0,  0,  0, 0, ...
  0,   2,   1,   0,  0,  0, 0, ...
  0,   6,   3,   1,  0,  0, 0, ...
  0,  24,  12,   6,  1,  0, 0, ...
  0, 120,  60,  35, 10,  1, 0, ...
  0, 720, 360, 226, 85, 15, 1, ...

Links

Crossrefs

Columns k=0-3 give: A000007, A000142, (-1)^n * A105752(n), A357828.
Cf. A357293.

Programs

  • PARI
    T(n, k) = sum(j=0, n, abs(stirling(n, k*j, 1)));
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (-log(1-x+x*O(x^n)))^(k*j)/(k*j)!), n));
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Pochhammer(w^j, n)))/k);

Formula

For k > 0, e.g.f. of column k: Sum_{j>=0} (-log(1-x))^(k*j)/(k*j)!.
For k > 0, T(n,k) = ( Sum_{j=0..k-1} (w^j)_n )/k, where (x)_n is the Pochhammer symbol and w = exp(2*Pi*i/k).

A357830 a(n) = Sum_{k=0..floor((n-2)/3)} |Stirling1(n,3*k+2)|.

Original entry on oeis.org

0, 0, 1, 3, 11, 51, 289, 1939, 15029, 132069, 1296771, 14063721, 166897059, 2150579067, 29895590361, 445871456667, 7100686041813, 120249378265653, 2157637558311963, 40887284144179473, 815949872494416387, 17103401793743095467, 375692072337527815233
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357828, A357829.

Programs

  • Maple
    f:= proc(n) local k;  add(abs(Stirling1(n,3*k+2)), k=0..(n-2)/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 12 2024
  • Mathematica
    Table[Sum[Abs[StirlingS1[n,3k+2]],{k,0,Floor[(n-2)/3]}],{n,0,30}] (* Harvey P. Dale, Jan 12 2024 *)
  • PARI
    a(n) = sum(k=0, (n-2)\3, abs(stirling(n, 3*k+2, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, (-log(1-x))^(3*k+2)/(3*k+2)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+w*Pochhammer(w, n)+w^2*Pochhammer(w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(-log(1-x)).
a(n) = ( (1)_n + w * (w)_n + w^2 * (w^2)_n )/3, where (x)_n is the Pochhammer symbol.

A357829 a(n) = Sum_{k=0..floor((n-1)/3)} |Stirling1(n,3*k+1)|.

Original entry on oeis.org

0, 1, 1, 2, 7, 34, 205, 1456, 11837, 108150, 1096011, 12196128, 147814359, 1938062490, 27333191613, 412614191808, 6638401596645, 113398127795862, 2049808094564139, 39091473755006400, 784404343854767727, 16520634668922810426, 364400233756422553053
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357828, A357830.

Programs

  • PARI
    a(n) = sum(k=0, (n-1)\3, abs(stirling(n, 3*k+1, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, (-log(1-x))^(3*k+1)/(3*k+1)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+w^2*Pochhammer(w, n)+w*Pochhammer(w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(-log(1-x)).
a(n) = ( (1)_n + w^2 * (w)_n + w * (w^2)_n )/3, where (x)_n is the Pochhammer symbol.

A384836 a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n,4*k)|.

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6770, 67320, 724550, 8427650, 105615500, 1420941600, 20448793300, 313670857500, 5111631733000, 88224807112000, 1608190674259000, 30879323250633000, 623074177992110000, 13182400475167560000, 291842125111122170000, 6748135840840046510000
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2025

Keywords

Crossrefs

Cf. A000142, A001710, A357828, A384837.
Cf. A000110, A024430, A143815, A365525, A365528.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, 4*k]], {k, 0, Floor[n/4]}], {n, 0, 30}]
  • PARI
    a(n) = sum(k=0, n\4, abs(stirling(n, 4*k, 1))); \\ Michel Marcus, Jun 10 2025

Formula

a(n) ~ n!/4.

A384837 a(n) = Sum_{k=0..floor(n/5)} |Stirling1(n,5*k)|.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269326, 3416985, 45997655, 657262606, 9959178229, 159758917956, 2707741441460, 48389066401764, 909877831207125, 17965423056654249, 371766710374672096, 8047954162682335066, 181941000229690525197, 4288430328840863166236, 105226297616943093770399
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2025

Keywords

Crossrefs

Cf. A000142, A001710, A357828, A384836.
Cf. A000110, A024430, A143815, A365525, A365528.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, 5*k]], {k, 0, Floor[n/5]}], {n, 0, 30}]
  • PARI
    a(n) = sum(k=0, n\5, abs(stirling(n, 5*k, 1))); \\ Michel Marcus, Jun 10 2025

Formula

a(n) ~ n!/5.
Showing 1-5 of 5 results.

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