A101853 -id:A101853 - cp's OEIS frontend

A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 13, 0, 1, 5, 18, 37, 47, 24, 0, 1, 6, 25, 64, 111, 110, 48, 0, 1, 7, 33, 100, 215, 303, 258, 86, 0, 1, 8, 42, 146, 370, 660, 804, 568, 160, 0, 1, 9, 52, 203, 588, 1251, 1938, 2022, 1237, 282, 0

Offset: 0

Alois P. Heinz, Mar 11 2015

A(n,k) is the number of partitions of n when parts i are of k*i kinds. A(2,2) = 7: [2a], [2b], [2c], [2d], [1a,1a], [1a,1b], [1b,1b].

			Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6,     7, ...
  0,  3,   7,   12,   18,    25,    33,    42, ...
  0,  6,  18,   37,   64,   100,   146,   203, ...
  0, 13,  47,  111,  215,   370,   588,   882, ...
  0, 24, 110,  303,  660,  1251,  2160,  3486, ...
  0, 48, 258,  804, 1938,  4005,  7459, 12880, ...
  0, 86, 568, 2022, 5400, 12150, 24354, 44885, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000219, A161870, A255610, A255611, A255612, A255613, A255614, A193427, A316461, A316462.
Rows n=0-3 give: A000012, A001477, A055998, A101853.
Main diagonal gives A255672.
Antidiagonal sums give A299166.
Cf. A144064, A257673.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2016, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j*k).

T(n,k) = Sum_{i=0..k} C(k,i) * A257673(n,k-i).

A101854 a(n) = n(n+1)(n^2 + 21*n + 50)/24.

Original entry on oeis.org

6, 24, 61, 125, 225, 371, 574, 846, 1200, 1650, 2211, 2899, 3731, 4725, 5900, 7276, 8874, 10716, 12825, 15225, 17941, 20999, 24426, 28250, 32500, 37206, 42399, 48111, 54375, 61225, 68696, 76824, 85646, 95200, 105525, 116661, 128649, 141531, 155350

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004

5th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

5th row of the array shown in A101853.

Partial sums of A101853.

Table[25 n/12+(71n^2)/24+(11n^3)/12+n^4/24,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{6,24,61,125,225},40] (* Harvey P. Dale, Nov 05 2011 *)

G.f.: x*(6 - 6*x + x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 5. - Harvey P. Dale, Nov 05 2011
E.g.f.: exp(x)*x*(144 + 144*x + 28*x^2 + x^3)/24. - Stefano Spezia, Oct 14 2022

Formula moved to be the definition by Eric M. Schmidt, Dec 12 2013

A101855 a(n) = n(n+1)(n+2)(n+4)(n+23)/120.

Original entry on oeis.org

6, 30, 91, 216, 441, 812, 1386, 2232, 3432, 5082, 7293, 10192, 13923, 18648, 24548, 31824, 40698, 51414, 64239, 79464, 97405, 118404, 142830, 171080, 203580, 240786, 283185, 331296, 385671, 446896, 515592, 592416, 678062, 773262, 878787, 995448

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004

6th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 3: 1,4,1. 6th row of the array shown in A101853. Partial sums of A101854.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Table[n(n+1)(n+2)(n+4)(n+23)/120,{n,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{6,30,91,216,441,812},40](* Harvey P. Dale, Feb 07 2013 *)

G.f.: x*(6-6*x+x^2) / (x-1)^6. - R. J. Mathar, Dec 06 2011

a(1)=6, a(2)=30, a(3)=91, a(4)=216, a(5)=441, a(6)=812, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Feb 07 2013

cp's OEIS Frontend

A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

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A101854 a(n) = n(n+1)(n^2 + 21*n + 50)/24.

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A101855 a(n) = n(n+1)(n+2)(n+4)(n+23)/120.

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cp's OEIS Frontend

A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A101854 a(n) = n*(n+1)*(n^2 + 21*n + 50)/24.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A101855 a(n) = n*(n+1)*(n+2)*(n+4)*(n+23)/120.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A101854 a(n) = n(n+1)(n^2 + 21*n + 50)/24.

A101855 a(n) = n(n+1)(n+2)(n+4)(n+23)/120.