A006005 -id:A006005 - cp's OEIS frontend

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

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).

A366011 G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} prime(k) * A(x)^k.

Original entry on oeis.org

1, -3, 13, -67, 379, -2267, 14065, -89515, 580379, -3815731, 25356051, -169898467, 1145825123, -7767073859, 52858784539, -360823702851, 2468606626813, -16915880702563, 116028193869805, -796200367684945, 5463239318586601, -37465765826805457, 256664142145450777, -1755608165010431139

Offset: 1

Ilya Gutkovskiy, Sep 25 2023

sign

Reversion of g.f. for odd prime numbers together with 1.

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A006005, A007296.

nmax = 24; A[] = 0; Do[A[x] = x - Sum[Prime[k] A[x]^k, {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
nmax = 24; CoefficientList[InverseSeries[Series[x + Sum[Prime[k] x^k, {k, 2, nmax}], {x, 0, nmax}], x], x] // Rest

cp's OEIS Frontend

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

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