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 1-6 of 6 results.

A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 4, 18, 48, 480, 1440, 20160, 120960, 1451520, 7257600, 239500800, 958003200, 24908083200, 348713164800, 6538371840000, 41845579776000, 2134124568576000, 12804747411456000, 729870602452992000, 9731608032706560000, 204363768686837760000, 2248001455555215360000, 206816133911079813120000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 22 2018

Keywords

Links

Crossrefs

Column 1 of A338805.
Cf. A000005, A028342, A038048, A318250, A338814.

Programs

  • Mathematica
    Table[(n - 1)! DivisorSigma[0, n], {n, 1, 24}]
nmax = 24; Rest[CoefficientList[Series[Sum[Sum[x^(j k)/(j k), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
  • PARI
    a(n) = (n-1)!*numdiv(n); \\ Michel Marcus, Aug 22 2018

Formula

E.g.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^(1/k)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A028342.
a(p^k) = (k + 1)*(p^k - 1)!, where p is a prime.

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)!.

A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1+x^j)^(u/j).

Original entry on oeis.org

1, 0, 1, 4, 0, 1, -6, 16, 0, 1, 48, -30, 40, 0, 1, 0, 448, -90, 80, 0, 1, 1440, -840, 2128, -210, 140, 0, 1, -10080, 23532, -6720, 7168, -420, 224, 0, 1, 120960, -127008, 177868, -30240, 19488, -756, 336, 0, 1, 0, 2191104, -1018080, 892540, -100800, 45696, -1260, 480, 0, 1
Offset: 1

Views

Author

Seiichi Manyama, Nov 10 2020

Keywords

Comments

Also the Bell transform of A338814.

Examples

			exp(Sum_{n>0} u*A048272(n)*x^n/n) = 1 + u*x + u^2*x^2/2! + (4*u+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       0,     1;
       4,     0,     1;
      -6,    16,     0,    1;
      48,   -30,    40,    0,    1;
       0,   448,   -90,   80,    0,   1;
    1440,  -840,  2128, -210,  140,   0, 1;
  -10080, 23532, -6720, 7168, -420, 224, 0, 1;
  ...

Links

Crossrefs

Column k=1 gives A338814.
Row sums give A168243.
Cf. A048272, A075525, A338805.

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
  • PARI
    {T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1+x^j+x*O(x^n))^(u/j)), n), k)}
  • PARI
    a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
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*A048272(n)*x^n/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} A048272(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} A048272(i_j)/i_j.

A338864 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(x^j/(1 - x^j)) )^u.

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 72, 96, 24, 1, 240, 840, 360, 40, 1, 2880, 7200, 4920, 960, 60, 1, 10080, 70560, 65520, 19320, 2100, 84, 1, 161280, 745920, 887040, 362880, 58800, 4032, 112, 1, 1088640, 7983360, 12640320, 6652800, 1481760, 150192, 7056, 144, 1
Offset: 1

Views

Author

Seiichi Manyama, Nov 13 2020

Keywords

Comments

Also the Bell transform of A323295.

Examples

			exp(Sum_{n>0} u*d(n)*x^n) = 1 + u*x + (4*u+u^2)*x^2/2! + (12*u+12*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       4,      1;
      12,     12,      1;
      72,     96,     24,      1;
     240,    840,    360,     40,     1;
    2880,   7200,   4920,    960,    60,    1;
   10080,  70560,  65520,  19320,  2100,   84,   1;
  161280, 745920, 887040, 362880, 58800, 4032, 112, 1;
  ...

Links

Crossrefs

Column k=1..2 give A323295, (n!/2) * A055507(n-1).
Rows sum give A294363.
Cf. A000005 (d(n)), A338805, A338865, A338870.

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[0, 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(x^j/(1-x^j+x*O(x^n)))^u), n), k)}
  • PARI
    a(n) = if(n<1, 0, n!*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

E.g.f.: exp(Sum_{n>0} u*d(n)*x^n), where d(n) is the number of divisors of 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*d(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} d(i_j).

A338810 a(n) = (n!/2) * Sum_{k=1..n-1} d(k)*d(n-k)/(k*(n-k)), where d(n) is the number of divisors of n.

Original entry on oeis.org

0, 1, 6, 28, 170, 988, 7896, 60492, 555264, 5819904, 61776000, 725950080, 9894493440, 137963243520, 1875645434880, 33258387456000, 528975488563200, 9760969019289600, 175565885864140800, 3608256006957772800, 72367669059194880000, 1745463407406243840000
Offset: 1

Views

Author

Seiichi Manyama, Nov 10 2020

Keywords

Crossrefs

Column 2 of A338805.
Cf. A000005, A055507, A059356.

Programs

  • Mathematica
    a[n_] := (n - 1)! * Sum[DivisorSigma[0, k] * DivisorSigma[0, n - k]/k, {k, 1, n - 1} ]; Array[a, 22] (* Amiram Eldar, Nov 10 2020 *)
  • PARI
    {a(n)= n!*sum(k=1, n-1, numdiv(k)*numdiv(n-k)/(k*(n-k)))/2}
  • PARI
    {a(n)= (n-1)!*sum(k=1, n-1, numdiv(k)*numdiv(n-k)/k)}
  • PARI
    {a(n) = my(u='u); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-u/k)), n), 2)}

Formula

a(n) = (n-1)! * Sum_{k=1..n-1} d(k)*d(n-k)/k.

A338811 a(n) = (n!/6) * Sum_{i,j,k > 0 and i+j+k=n} d(i)*d(j)*d(k)/(i*j*k), where d(n) is the number of divisors of n.

Original entry on oeis.org

0, 0, 1, 12, 100, 870, 7588, 73808, 764524, 8448120, 103816944, 1334764728, 18483356736, 274780501632, 4371694872192, 71815113008640, 1282261138007040, 23828058693642240, 468231649812725760, 9599857257164820480, 205863214718290636800, 4646428416182168985600
Offset: 1

Views

Author

Seiichi Manyama, Nov 10 2020

Keywords

Crossrefs

Column 3 of A338805.
Cf. A000005, A059357.

Programs

  • PARI
    {a(n) = my(u='u); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-u/k)), n), 3)}
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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