login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A263234 Triangle read by rows: T(n,k) is the number of partitions of n having k triangular number parts (0<=k<=n). 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

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;

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, May 25 2018 *)

CROSSREFS

Cf. A000041, A000217, A225044, A263235.

Sequence in context: A291969 A321434 A103919 * A264394 A283310 A035445

Adjacent sequences:  A263231 A263232 A263233 * A263235 A263236 A263237

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Nov 12 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 00:41 EST 2019. Contains 329816 sequences. (Running on oeis4.)