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 1-3 of 3 results.

A222061 Triangle read by rows: coefficients of second-order hypergeometric-harmonic polynomials.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 15, 26, 0, 1, 35, 156, 154, 0, 1, 75, 650, 1540, 1044, 0, 1, 155, 2340, 10010, 15660, 8028, 0, 1, 315, 7826, 53900, 146160, 168588, 69264, 0, 1, 635, 25116, 261954, 1096200, 2135448, 1939392, 663696, 0, 1, 1275, 78650, 1196580, 7256844
Offset: 0

Views

Author

N. J. A. Sloane, Feb 08 2013

Keywords

Examples

			Triangle begins:
  0
  0 1
  0 1 5
  0 1 15 26
  0 1 35 156 154
  0 1 75 650 1540 1044
  ....

Links

Crossrefs

Cf. A222057-A222064. Row sums are in A222062.

Programs

  • Mathematica
    H2[k_] := (k+1) (HarmonicNumber[k+1] - 1);
T[n_, k_] := StirlingS2[n, k] k! H2[k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 01 2018 *)
  • PARI
    hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2)};
\\ Michel Marcus, Feb 09 2013

Formula

T(n,k) = A008277(n,k)*A000142(k)*H2(k) where H2(k) is defined by g.f.:- log(1-x)/(1-x)^2. - Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013

A222062 a(n) = n-th second-order hypergeometric-harmonic number.

Original entry on oeis.org

0, 1, 6, 42, 346, 3310, 36194, 446054, 6122442, 92668302, 1533812722, 27565147126, 534621745178, 11131104732254, 247646911102530, 5863652049020358, 147225092025474154, 3907328980930705966, 109297865960259305618, 3214017757399205062550, 99121172016580291190970
Offset: 0

Views

Author

N. J. A. Sloane, Feb 08 2013

Keywords

Links

Crossrefs

Cf. A222057-A222064. Row sums of A222061.

Programs

  • PARI
    hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
a(n) = {sum(k=0, n, k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2));}
\\ Michel Marcus, Feb 09 2013

Formula

a(n) = Sum_{k=0..n} A008277(n,k)*A000142(k)*H2(k) where H2(k) is defined by g.f.: - log(1-x)/(1-x)^2. - Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013

A222063 Triangle read by rows: coefficients of third-order hypergeometric-harmonic polynomials.

Original entry on oeis.org

0, 0, 1, 0, 1, 7, 0, 1, 21, 47, 0, 1, 49, 282, 342, 0, 1, 105, 1175, 3420, 2754, 0, 1, 217, 4230, 22230, 41310, 24552, 0, 1, 441, 14147, 119700, 385560, 515592, 241128, 0, 1, 889, 45402, 581742, 2891700, 6530832, 6751584, 2592720, 0, 1, 1785, 142175, 2657340
Offset: 0

Views

Author

N. J. A. Sloane, Feb 08 2013

Keywords

Examples

			Triangle begins:
0
0 1
0 1 7
0 1 21 47
0 1 49 282 342
0 1 105 1175 3420 2754
....

Links

Crossrefs

Cf. A222057-A222064. Row sums give A222064.

Programs

  • PARI
    hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,3)};
\\ Michel Marcus, Feb 09 2013

Formula

T(n,k) = A008277(n,k)*A000142(k)*H3(k) where H3(k) is defined by g.f.:- log(1-x)/(1-x)^3. - Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013
Showing 1-3 of 3 results.

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

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