A263235 -id:A263235 - cp's OEIS frontend

A263234 Triangle read by rows: T(n,k) is the number of partitions of n having k triangular number parts (0<=k<=n).

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 1, 2, 4, 2, 4, 0, 2, 0, 1, 4, 4, 5, 2, 4, 0, 2, 0, 1, 4, 6, 5, 6, 2, 4, 0, 2, 0, 1, 5, 9, 8, 5, 6, 2, 4, 0, 2, 0, 1, 6, 10, 11, 9, 5, 6, 2, 4, 0, 2, 0, 1, 9, 13, 13, 12, 10, 5, 6, 2, 4, 0, 2, 0, 1

Offset: 0

Emeric Deutsch, Nov 12 2015

The triangular numbers are i(i+1)/2 (i=0,1,2,3,...) (A000217).
Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,0) = A225044(n).
Sum_{k=0..n} k*T(n,k) = A263235(n) = total number of triangular number parts in all partitions of n.

			T(6,2) = 4 because we have [4,1,1], [3,3], [3,2,1], and [2,2,1,1] (the partitions of 6 that have 2 triangular number parts).
Triangle starts:
  1;
  0,1;
  1,0,1;
  0,2,0,1;
  2,0,2,0,1;
  1,3,0,2,0,1;

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

Cf. A000041, A000217, A225044, A263235.

h := proc (i) options operator, arrow: (1/2)*i*(i+1) end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

max = 15; h[i_] = i*(i + 1)/2; P = Product[(1 - x^h[i])/((1 - x^i)*(1 - t*x^h[i])), {i, 1, max}] + O[x]^max;
CoefficientList[#, t]& /@ CoefficientList[P, x] // Flatten (* Jean-François Alcover, May 25 2018 *)

G.f.: Product_{i>0} (1-x^h(i))/((1-x^i)*(1-t*x^h(i))), where h(i) = i*(i+1)/2.

A309536 Total number of triangular numbers in all compositions of n.

Original entry on oeis.org

0, 1, 2, 6, 14, 33, 77, 174, 389, 860, 1885, 4098, 8853, 19020, 40668, 86593, 183698, 388421, 818892, 1721884, 3611968, 7560337, 15793474, 32932549, 68556300, 142495004, 295754816, 613039248, 1269137729, 2624393922, 5421024773, 11186523404, 23061994524

Offset: 0

Alois P. Heinz, Aug 06 2019

nonn

			a(4) = 14: (1)(1)(1)(1), 2(1)(1), (1)2(1), (1)(1)2, 22, (3)(1), (1)(3), 4.

Alois P. Heinz, Table of n, a(n) for n = 0..3312

Cf. A000217, A102291, A263235.

a:= proc(n) option remember; add(a(n-j)+
      `if`(issqr(8*j+1), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)) - 1)*((1 - x)^2/(1 - 2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)

G.f.: Sum_{k>=1} x^(k*(k+1)/2)*(1-x)^2/(1-2*x)^2.

a(n) ~ c * 2^n * n, where c = 0.1604081401637884665734606925563573585565153844... - Vaclav Kotesovec, Aug 18 2019

cp's OEIS Frontend

A263234 Triangle read by rows: T(n,k) is the number of partitions of n having k triangular number parts (0<=k<=n).

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