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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)