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.

A007042 Left diagonal of partition triangle A047812.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 40, 54, 75, 99, 133, 174, 229, 295, 383, 488, 625, 790, 1000, 1253, 1573, 1956, 2434, 3008, 3716, 4563, 5602, 6840, 8347, 10141, 12308, 14881, 17975, 21635, 26013, 31183, 37336, 44581, 53172, 63259, 75173, 89132, 105556, 124752
Offset: 1

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Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010
Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012
This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.
Column k = 2 of A081719.

Programs

  • Julia
    using Nemo
function A007042List(len)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(-1, len+2, z)
    [coeff(e, j) - 2 for j in 2:len+1] end
A007042List(45) |> println # Peter Luschny, May 30 2020
  • Mathematica
    f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* Clark Kimberling, Mar 13 2012 *)
a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* Jean-François Alcover, Jan 28 2015, after M. F. Hasler *)
  • PARI
    A007042(n)=numbpart(n+1)-2  \\ M. F. Hasler, Apr 12 2012

Formula

a(n) = A000041(n+1) - 2. - Vladeta Jovovic, Oct 06 2001

Extensions

More terms from James Sellers
Name edited by Petros Hadjicostas, May 31 2020

A022818 Square array read by antidiagonals: A(n,k) = number of terms in the n-th derivative of a function composed with itself k times (n, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 13, 7, 1, 1, 6, 15, 26, 23, 11, 1, 1, 7, 21, 45, 55, 44, 15, 1, 1, 8, 28, 71, 110, 121, 74, 22, 1, 1, 9, 36, 105, 196, 271, 237, 129, 30, 1, 1, 10, 45, 148, 322, 532, 599, 468, 210, 42, 1, 1, 11, 55, 201, 498, 952, 1301, 1309, 867, 345, 56, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Square array A(n,k) (with rows n >= 1 and columns k >= 1) begins:
  1,  1,   1,   1,    1,    1,    1,     1, ...
  1,  2,   3,   4,    5,    6,    7,     8, ...
  1,  3,   6,  10,   15,   21,   28,    36, ...
  1,  5,  13,  26,   45,   71,  105,   148, ...
  1,  7,  23,  55,  110,  196,  322,   498, ...
  1, 11,  44, 121,  271,  532,  952,  1590, ...
  1, 15,  74, 237,  599, 1301, 2541,  4586, ...
  1, 22, 129, 468, 1309, 3101, 6539, 12644, ...
  ...

References

  • Winston C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Links

Crossrefs

Rows n=1-10 give: A000012, A000027, A000217, A008778, A022815, A022816, A022817, A215626, A215627, A215628.
Columns k=1-10 give: A000012, A000041, A022811, A022812, A022813, A022814, A024207, A024208, A024209, A024210.
Main diagonal gives: A192435.
Cf. A008284, A081719.

Programs

  • Maple
    A:= proc(n, k) option remember;
      `if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
    end:
b:= proc(n, i, k) option remember; `if`(nAlois P. Heinz, Aug 18 2012
# second Maple program:
b:= proc(n, i, l, k) option remember; `if`(k=0,
      `if`(n<2, 1, 0), `if`(n=0 or i=1, b(l+n$2, 0, k-1),
         b(n, i-1, l, k) +b(n-i, min(n-i, i), l+1, k)))
    end:
A:= (n, k)->  b(n$2, 0, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jul 19 2018
  • Mathematica
    a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]]; b[n_, i_, k_] := b[n, i, k] = If[n < k, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n-i*j, i-1, k-j], {j, 0, Min[n/i, k]}]]]]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)
  • PARI
    P(n, k) = #partitions(n-k, k); /* A008284 */
tabl(nn) = {M = matrix(nn, nn, n, k, 0); for(n=1, nn, M[n, 1] = 1; ); for(n=1, nn, for(k=2, nn, M[n, k] = sum(s=1, n, P(n, s)*M[s, k-1]))); for (n=1, nn, for (k=1, nn, print1(M[n, k], ", "); ); print(); ); } \\ Petros Hadjicostas, May 30 2020

Formula

From Petros Hadjicostas, May 30 2020: (Start)
A(n,k) = Sum_{s=1..n} A008284(n,s)*A(s,k-1) for n >= 1 and k >= 2 with A(n,1) = 1 for n >= 1.
A(n,k) = Sum_{s=1..n} binomial(k,s-1)*A081719(n-1,s-1) for n, k >= 1. (End)

Extensions

Edited by Alois P. Heinz, Aug 18 2012

A039809 For n > 1, a(n) doubles under the transform T, where Ta is the matrix product of partition triangle A008284 with a, with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 12, 32, 83, 223, 594, 1600, 4297, 11589, 31216, 84212, 227091, 612712, 1652913, 4459962, 12033405, 32469682, 87611105, 236402465, 637884103, 1721218224, 4644392797, 12532091909, 33815653370, 91245738923
Offset: 1

Views

Author

Christian G. Bower, Feb 15 1999

Keywords

Examples

			So a(7) = T(7,1)*a(1) + T(7,2)*a(2) + ... + T(7,6)*a(6) = 1*1 + 3*1 + 4*2 + 3*5 + 2*12 + 1*32 = 1 + 3 + 8 + 15 + 24 + 32 = 83, where T(n,k) = A008284(n,k).

Crossrefs

Cf. A008284, A039808, A081719.

Programs

  • PARI
    P(n, k) = #partitions(n-k, k); /* A008284 */
lista(nn) = {my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n] = sum(i=1, n-1, P(n,i)*a[i])); a;} \\ Petros Hadjicostas, May 30 2020

Formula

a(1) = 1 and a(n) = Sum_{i=1..n-1} A008284(n, i)*a(i) for n >= 2 (because 2*a(n) = Sum_{i=1..n} A008284(n,i)*a(i) for n >= 2).
a(n+1) = Sum_{k=0..n} A081719(n,k). - Philippe Deléham, Sep 30 2006
G.f.: (1/2) * ( x + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - x^j) ). - Ilya Gutkovskiy, Jul 22 2021

Extensions

Various sections edited by Petros Hadjicostas, May 30 2020
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