A316706 -id:A316706 - cp's OEIS frontend

A367736 a(0) = 1; for n > 0, a(n) is the coefficient of x^a(n-1) in the expansion of Product_{k=0..n-1} (x^a(k) + 1 + 1/x^a(k)).

1, 1, 2, 4, 6, 11, 19, 32, 58, 97, 163, 290, 501, 856, 1483, 2561, 4424, 7652, 13273, 23024, 39784, 69001, 119614, 207042, 358746, 621117, 1075865, 1864050, 3227724, 5590548, 9682402, 16770033, 29049713, 50310453, 87142439, 150939346, 261424583, 452810957

Offset: 0

Ilya Gutkovskiy, Jan 24 2024

Cf. A063890, A265853, A316706.

a[0] = 1; a[n_] := a[n] = Coefficient[Product[x^a[k] + 1 + 1/x^a[k], {k, 0, n - 1}], x, a[n - 1]]; Table[a[n], {n, 0, 28}]

from itertools import islice
from collections import Counter
def A367736_gen(): # generator of terms
    c, b = {0:1}, 1
    while True:
        yield b
        d = Counter(c)
        for k in c:
            e = c[k]
            d[k+b] += e
            d[k-b] += e
        c = d
        b = c[b]
A367736_list = list(islice(A367736_gen(),20)) # Chai Wah Wu, Feb 05 2024

a(29)-a(37) from Chai Wah Wu, Feb 05 2024

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 11, 25, 56, 129, 313, 748, 1831, 4584, 11581, 29555, 76398, 199191, 522746, 1382158, 3676072, 9828631, 26412174, 71310248, 193346941, 526302232, 1437895915, 3941671019, 10839118898, 29893768219, 82671240995, 229213877639, 637049969282

Offset: 0

Ilya Gutkovskiy, Jan 24 2024

nonn

Cf. A000217, A316706, A351002, A369344, A369433.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
     `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+b(n, i-1)+
        b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 24 2024

Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, n], {n, 0, 32}]

A369517 a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 9, 20, 43, 85, 183, 394, 1010, 2254, 5589, 12383, 31226, 71153, 182382, 426055, 1105686, 2615167, 6906858, 16607500, 44276140, 107836782, 290059089, 715361182, 1937639649, 4829754357, 13160903826, 33112002835, 90800047879

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A316706, A369358, A369434, A369438.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(n, i-1)+
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> b(n^4, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 25 2024

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-1] + b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30];
a[n_] := b[n^4, n];
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Feb 14 2025, after Alois P. Heinz *)

a(34)-a(37) from Alois P. Heinz, Jan 25 2024

cp's OEIS Frontend

A367736 a(0) = 1; for n > 0, a(n) is the coefficient of x^a(n-1) in the expansion of Product_{k=0..n-1} (x^a(k) + 1 + 1/x^a(k)).

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A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

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A369517 a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

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cp's OEIS Frontend

A367736 a(0) = 1; for n > 0, a(n) is the coefficient of x^a(n-1) in the expansion of Product_{k=0..n-1} (x^a(k) + 1 + 1/x^a(k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Extensions

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A369517 a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Extensions

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).