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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A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*d(n)*x^n/n!), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 3, 20, 12, 1, 2, 55, 80, 20, 1, 4, 142, 405, 220, 30, 1, 2, 322, 1792, 1785, 490, 42, 1, 4, 779, 7224, 12152, 5810, 952, 56, 1, 3, 1608, 27323, 73920, 56532, 15498, 1680, 72, 1, 4, 3894, 99690, 414815, 482160, 204204, 35910, 2760, 90, 1
Offset: 1

Views

Author

Seiichi Manyama, Nov 13 2020

Keywords

Comments

Also the Bell transform of A000005.

Examples

			exp(Sum_{n>0} u*d(n)*x^n/n!) = 1 + u*x + (2*u+u^2)*x^2/2! + (2*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
  1;
  2,   1;
  2,   6,    1;
  3,  20,   12,     1;
  2,  55,   80,    20,    1;
  4, 142,  405,   220,   30,   1;
  2, 322, 1792,  1785,  490,  42,  1;
  4, 779, 7224, 12152, 5810, 952, 56, 1;
  ...

Links

Crossrefs

Column k=1..2 give A000005, A328681(n-1).
Row sums give A295739.
Cf. A338805, A338864, A338871.

Programs

  • Mathematica
    T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[0, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)
  • PARI
    a(n) = if(n<1, 0, numdiv(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

Formula

T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * Sum_{k=1..n} binomial(n-1,k-1)*d(k)*T(n-k; u), T(0; u) = 1.
T(n,k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} d(i_j)/(i_j)!.

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

Original entry on oeis.org

1, 6, 1, 24, 18, 1, 168, 204, 36, 1, 720, 2280, 780, 60, 1, 8640, 25200, 14400, 2100, 90, 1, 40320, 292320, 252000, 58800, 4620, 126, 1, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1
Offset: 1

Views

Author

Seiichi Manyama, Nov 13 2020

Keywords

Examples

			exp(Sum_{n>0} u*sigma(n)*x^n) = 1 + u*x + (6*u+u^2)*x^2/2! + (24*u+18*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       6,       1;
      24,      18,       1;
     168,     204,      36,       1;
     720,    2280,     780,      60,      1;
    8640,   25200,   14400,    2100,     90,    1;
   40320,  292320,  252000,   58800,   4620,  126,   1;
  604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
  ...

Links

Crossrefs

Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).
Rows sum give A294361.
Cf. A000203 (sigma(n)), A008298, A338864, A338871.

Programs

  • Mathematica
    T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * j! * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)
  • PARI
    {T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, exp(j*x^j/(1-x^j+x*O(x^n)))^u), n), k)}
  • PARI
    a(n) = if(n<1, 0, n!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

Formula

E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^n).
T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} k*sigma(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n,k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j).
Showing 1-2 of 2 results.

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