A131823 Triangle read by rows of [2^(n+1) - (n+1)] terms where row n has the g.f.: Product_{i=0..n-1} (1 + x^(2^i))^(n-i) for n>0, starting with a '1' in row 0.

1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 5, 7, 8, 8, 8, 8, 7, 5, 3, 1, 1, 4, 9, 16, 24, 32, 40, 48, 55, 60, 63, 64, 64, 64, 64, 64, 63, 60, 55, 48, 40, 32, 24, 16, 9, 4, 1, 1, 5, 14, 30, 54, 86, 126, 174, 229, 289, 352, 416, 480, 544, 608, 672, 735, 795, 850, 898, 938, 970, 994, 1010, 1019

Offset

0,5

Comments

Generating rule: Start with a single '1' in row 0; let S(n) denote the initial [2^(n+1) - (n+1)] terms of the partial sums of row n; generate row n+1 by concatenating the following: S(n), 2^[n*(n-1)/2] repeated (n-1) times and the terms of S(n) when read in reverse order.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1990

Jordan Stoyanov, Christophe Vignat, Non-Conventional Limits of Random Sequences Related to Partitions of Integers, arXiv:1901.04029 [math.PR], 2019. [See page 2 and appendix for additional properties of this sequence.}

C. Vignat, T. Wakhare, Finite generating functions for the sum of digits sequence, arXiv:1708.06479 [math.NT], 2017.

Tanay Wakhare, Christophe Vignat, Settling some sum suppositions, arXiv:1805.10569 [math.NT], 2018.

Tanay Wakhare, Christophe Vignat, Settling some sum suppositions, Acta Math. Hungar. (2018).

Formula

Row sums are 2^(n*(n+1)/2) for n>=0.

Example

Triangle begins:
1;
1,1;
1,2, 2, 2,1;
1,3,5,7,8, 8,8, 8,7,5,3,1;
1,4,9,16,24,32,40,48,55,60,63,64, 64,64,64, 64,63,60,55,48,40,32,24,16,9,4,1; ...
Illustrate the row g.f.s by:
(1+x)^2*(1+x^2) = g.f. of row 2: [1,2,2,2,1];
(1+x)^3*(1+x^2)^2*(1+x^4) = g.f. of row 3: [1,3,5,7,8,8,8,8,7,5,3,1];
(1+x)^4*(1+x^2)^3*(1+x^4)^2*(1+x^8) = g.f. of row 4.

Mathematica

Rest@ Flatten@ Array[{1}~Join~CoefficientList[Series[Product[(1 + x^(2^i))^(# - i), {i, 0, # - 1}], {x, 0, 2^(# + 1) - (# + 1)}], x] &, 5] (* Michael De Vlieger, Aug 21 2018 *)

Prog

(PARI) {T(n, k)=local(A=[1]); if(n==0, 1, for(i=0, n-1, A=concat(Vec((Polrev(A)+O(x^(#A+i)))/(1-x)), Vec(O(x^(#A))+Pol(Vec(Ser(A)/(1-x)))))); A[k+1])}
for(n=0, 6, for(k=0, 2^(n+1)-(n+2), print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A131824 (main diagonal).

Sequence in context: A286756 A193884 A128084 * A089722 A172356 A184948

Adjacent sequences: A131820 A131821 A131822 * A131824 A131825 A131826

Keyword

nonn,tabf

Author

Paul D. Hanna, Jul 19 2007

Status

approved