A319818 -id:A319818 - cp's OEIS frontend

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).

A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Cf. A000217, A007331, A010054, A053603, A053604, A319818, A340949, A340950, A340951, A340952, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^8, where theta_2() is the Jacobi theta function.

A341025 Number of partitions of n into 8 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 1, 1, 3, 1, 1, 2, 1, 2, 4, 2, 2, 2, 0, 2, 5, 2, 1, 5, 2, 2, 4, 3, 4, 2, 3, 4, 3, 5, 1, 8, 4, 3, 8, 1, 4, 7, 2, 5, 7, 5, 6, 6, 4, 4, 10, 6

Offset: 120

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A007331, A010054, A307597, A307598, A319818, A340953, A341021, A341022, A341023, A341024, A341026, A341027.

A341791 Number of partitions of n into 8 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 2, 3, 0, 3, 4, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 2, 4, 2, 1, 5, 3, 2, 6, 4, 2, 5, 3, 2, 5, 4, 2, 6, 4, 3, 5, 5, 2, 5, 5, 4, 6, 6, 3, 6, 6, 3, 5, 6, 3, 6, 8, 4, 5, 8, 4, 5, 8, 4, 5, 10

Offset: 8

Ilya Gutkovskiy, Feb 19 2021

nonn

Cf. A000292, A023533, A024692, A068980, A319818, A341774, A341775, A341776, A341777, A341778, A341792, A341793.

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.

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A341025 Number of partitions of n into 8 distinct nonzero triangular numbers.

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A341791 Number of partitions of n into 8 nonzero tetrahedral numbers.

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