A307898 -id:A307898 - cp's OEIS frontend

A307899 Expansion of 1/(1 + x * Sum_{k>=1} prime(k)*x^k).

1, 0, -2, -3, -1, 5, 10, 9, -4, -26, -43, -33, 35, 148, 219, 98, -316, -857, -983, 23, 2296, 4501, 3712, -2906, -14257, -21771, -10811, 28282, 81209, 97292, 7960, -207185, -431595, -386033, 219344, 1322141, 2134126, 1226554, -2443765, -7684081, -9726127, -1791806, 18712361, 41428590, 39753658

Offset: 0

Ilya Gutkovskiy, May 04 2019

sign

Alternating antidiagonal sums of square array, in which row m equals the m-fold convolution of primes with themselves.

Cf. A000040, A030017, A030018, A307898.

nmax = 44; CoefficientList[Series[1/(1 + x Sum[Prime[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Prime[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 44}]

Recurrence: a(n+1) = -Sum_{k=1..n} prime(k)*a(n-k).

A353156 a(0) = 1; a(n) = -Sum_{k=1..n} prime(k+1) * a(n-k).

Original entry on oeis.org

1, -3, 4, -4, 2, 6, -22, 46, -74, 86, -40, -120, 450, -958, 1506, -1694, 744, 2500, -9184, 19422, -30450, 34032, -14178, -52286, 188038, -394724, 615102, -681110, 268666, 1089974, -3847390, 8021030, -12426638, 13632728, -5063588, -22711916, 78708912, -162966020, 251005706

Offset: 0

Ilya Gutkovskiy, Apr 27 2022

sign

Cf. A006005, A030017, A030018, A065091, A292744, A300662, A307898, A307899, A346792.

a[0] = 1; a[n_] := a[n] = -Sum[Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[1/(1 + Sum[Prime[k + 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1 / (1 + Sum_{k>=1} prime(k+1) * x^k).

cp's OEIS Frontend

A307899 Expansion of 1/(1 + x * Sum_{k>=1} prime(k)*x^k).

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