A202261 -id:A202261 - cp's OEIS frontend

A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 3, 0, 1, 1, 7, 36, 7, 0, 1, 1, 15, 351, 785, 18, 0, 1, 1, 31, 3240, 56217, 26404, 51, 0, 1, 1, 63, 29403, 3695545, 18878418, 1235580, 155, 0, 1, 1, 127, 265356, 238085177, 12107973904, 11163952389, 74394425, 486, 0

Offset: 0

Alois P. Heinz, Jan 25 2012

A(n,k) is the number of partitions of n^(k+1) into n distinct parts <= 2*n^k.

			A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(2,2) = 3: {1,7}, {2,6}, {3,5}.
A(3,1) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
A(4,1) = 7: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
A(2,3) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
Square array A(n,k) begins:
  1,   1,      1,         1,            1, ...
  1,   1,      1,         1,            1, ...
  0,   1,      3,         7,           15, ...
  0,   3,     36,       351,         3240, ...
  0,   7,    785,     56217,      3695545, ...
  0,  18,  26404,  18878418,  12107973904, ...

Rows n=1-3 give: A000012, A000225, A026121.

Columns k=1-3 give: A202261, A186730, A185062.

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n^(k+1), 2*n^k, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);

$RecursionLimit = 10000; b[n_, i_, t_] := b[n, i, t] = If [i < t || n < t*(t+1)/2 || n > t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]]; A[0, ] = A[1, ] = 1; A[n_ /; n > 1, 0] = 0; A[n_, k_] := b[n^(k+1), 2*n^k, n]; Table[Print[ta = Table [A[n, d-n], {n, 0, d}]]; ta, {d, 0, 9}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A186730 Number of n-element subsets that can be chosen from {1,2,...,2*n^2} having element sum n^3.

Original entry on oeis.org

1, 1, 3, 36, 785, 26404, 1235580, 74394425, 5503963083, 484133307457, 49427802479445, 5750543362215131, 751453252349649771, 109016775078856564392, 17391089152542558703435, 3026419470005398093836960, 570632810506646981058828349, 115900277419940965862120360831

Offset: 0

Alois P. Heinz, Jan 21 2012

nonn

a(n) is the number of partitions of n^3 into n distinct parts <= 2*n^2.

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 3: {1,7}, {2,6}, {3,5}.

Column k=2 of A185282.

Cf. A202261, A204459.

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n^3, 2*n^2, n):
seq(a(n), n=0..12);

$RecursionLimit = 2000;
b[n_, i_, t_] := b[n, i, t] = If[it (2i-t+1)/2, 0, If[n==0, 1, b[n, i-1, t] + If[nJean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

A185062 Number of n-element subsets that can be chosen from {1,2,...,2*n^3} having element sum n^4.

Original entry on oeis.org

1, 1, 7, 351, 56217, 18878418, 11163952389, 10292468330630, 13703363417260677, 24932632800863823135, 59509756600504616529186, 180533923700628895521591343, 678854993880375551144618682344, 3100113915888360851262910882014885

Offset: 0

Alois P. Heinz, Jan 22 2012

nonn

a(n) is the number of partitions of n^4 into n distinct parts <= 2*n^3.

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.

Column k=3 of A185282.

Cf. A202261, A186730, A204459.

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n^4, 2*n^3, n):
seq(a(n), n=0..5);

$RecursionLimit = 10000;
b[n_, i_, t_] := b[n, i, t] =
     If[i < t || n < t(t+1)/2 || n > t(2i - t + 1)/2, 0,
     If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]];
a[n_] := b[n^4, 2 n^3, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).

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A186730 Number of n-element subsets that can be chosen from {1,2,...,2*n^2} having element sum n^3.

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A185062 Number of n-element subsets that can be chosen from {1,2,...,2*n^3} having element sum n^4.

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Maple

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