A206229 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^(n-k+1).

1, 1, 2, 8, 31, 124, 515, 2166, 9182, 39195, 168216, 725043, 3136223, 13606891, 59187790, 258034685, 1127137141, 4932071321, 21614913239, 94859273448, 416820578198, 1833626307670, 8074598332650, 35591081565244, 157013886785417, 693237405812328

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Examples

			Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] (1+x) = 1;
a(2) = [x^2] (1+x)^2*(1+x^2) = 2;
a(3) = [x^3] (1+x)^3*(1+x^2)^2*(1+x^3) = 8;
a(4) = [x^4] (1+x)^4*(1+x^2)^3*(1+x^3)^2*(1+x^4) = 31; ...
as illustrated below.
The coefficients in Product_{k=1..n} (1+x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 2,(2), 2, 1, 0, 0, 0, 0, 0, 0, 0 ...];
n=3: [1, 3, 5, (8), 10, 10, 10, 8, 5, 3, 1, 0 ...];
n=4: [1, 4, 9, 18, (31), 46, 64, 82, 96, 106, 110, 106 ...];
n=5: [1, 5, 14, 33, 68, (124), 210, 332, 492, 693, 931, ...];
n=6: [1, 6, 20, 54, 127, 266, (515), 934, 1597, 2602, ...];
n=7: [1, 7, 27, 82, 215, 502, 1078, (2166), 4109, 7428, ...];
n=8: [1, 8, 35, 118, 340, 870, 2038, 4454, (9182), 18020, ...];
n=9: [1, 9, 44, 163, 511, 1417, 3582, 8420, 18634,(39195), ...]; ...
where the coefficients in parenthesis start this sequence.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A206228.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)

PARI

{a(n)=polcoeff(prod(k=1,n,(1+x^k+x*O(x^n))^(n-k+1)),n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007844067631287560916216334645404240888403... and c = 0.1630284922981520921416997097273846855003438911417350833863798... - Vaclav Kotesovec, Aug 21 2018