A014347
Three-fold exponential convolution of primes with themselves.
Original entry on oeis.org
8, 36, 168, 786, 3660, 16866, 76752, 343914, 1514724, 6543066, 27699960, 114793386, 466078116, 1854554490, 7248419496, 27869755866, 105687130980, 395978680266, 1468425404328, 5396913313866, 19675676962308, 71219609783946, 256052236665192, 914773982356902
Offset: 0
-
b:= proc(n, k) option remember; `if`(k=1,
ithprime(n+1), add(b(j, floor(k/2))*
b(n-j, ceil(k/2))*binomial(n, j), j=0..n))
end:
a:= n-> b(n, 3):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 10 2018
-
b[n_, k_] := b[n, k] = If[k == 1, Prime[n + 1], Sum[b[j, Floor[k/2]] b[n - j, Ceiling[k/2]] Binomial[n, j], {j, 0, n}]];
a[n_] := b[n, 3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
A014352
Four-fold exponential convolution of primes with themselves.
Original entry on oeis.org
16, 96, 592, 3680, 22888, 141776, 872296, 5320160, 32116168, 191634736, 1128985544, 6560592320, 37577101096, 212032652336, 1178400630472, 6450745788064, 34795044655624, 185041871051312, 971039709861320, 5033044804735360, 25793494764933224, 130834363186542320
Offset: 0
-
b:= proc(n, k) option remember; `if`(k=1,
ithprime(n+1), add(b(j, floor(k/2))*
b(n-j, ceil(k/2))*binomial(n, j), j=0..n))
end:
a:= n-> b(n, 4):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 10 2018
-
b[n_, k_] := b[n, k] = If[k == 1, Prime[n + 1], Sum[b[j, Floor[k/2]] b[n - j, Ceiling[k/2]] Binomial[n, j], {j, 0, n}]];
a[n_] := b[n, 4];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
A333371
Exponential convolution of primorial numbers (A002110) with themselves.
Original entry on oeis.org
1, 4, 20, 132, 1116, 12420, 171300, 2884980, 56674380, 1289511300, 34769949060, 1063909626780, 37255008811020, 1470406699982220, 63114539746598340, 2936218980067393020, 150241360192861037100, 8497891914008911514100, 514514062115556069627060
Offset: 0
-
p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(p(i)*p(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 17 2020
-
primorial[n_] := Product[Prime[k], {k, 1, n}]; a[n_] := Sum[Binomial[n, k] primorial[k] primorial[n - k], {k, 0, n}]; Table[a[n], {n, 0, 18}]
A014346
Exponential convolution of primes with themselves (divided by 2).
Original entry on oeis.org
2, 6, 19, 59, 181, 541, 1583, 4455, 12213, 32113, 82631, 206709, 510681, 1245343, 3004575, 7200705, 17049021, 40140981, 93678375, 216274577, 496470625, 1128356231, 2544413119, 5704402931, 12712869673, 28191681427, 62282778949, 137195275297, 300840332783
Offset: 0
A300631
a(n) = n! * [x^n] (Sum_{k=0..n} prime(k+1)*x^k/k!)^n.
Original entry on oeis.org
1, 3, 38, 786, 22888, 857800, 39316464, 2130380560, 133222474368, 9443111340672, 748168002970880, 65520799156209408, 6284786657494483968, 655287035001111884800, 73792143714173551392768, 8925528145554323771934720, 1154065253662722209679572992, 158849709577131169400652988416
Offset: 0
The table of coefficients of x^k in expansion of e.g.f. (Sum_{k>=0} prime(k+1)*x^k/k!)^n begins:
n = 0: (1), 0, 0, 0, 0, 0, ... (A000007)
n = 1: 2, (3), 5, 7, 11, 13, ... (A000040, with offset 0)
n = 2: 4, 12, (38), 118, 362, 1082, ... (A014345)
n = 3: 8, 36, 168, (786), 3660, 16866, ... (A014347)
n = 4: 16, 96, 592, 3680, (22888), 141776, ... (A014352)
n = 5: 32, 240, 1840, 14240, 110560, (857800), ...
-
b:= proc(n, k) option remember; `if`(k=1, ithprime(n+1), add(
b(j, floor(k/2))*b(n-j, ceil(k/2))*binomial(n, j), j=0..n))
end:
a:= n-> `if`(n=0, 1, b(n$2)):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 10 2018
-
Table[n! SeriesCoefficient[Sum[Prime[k + 1] x^k/k!, {k, 0, n}]^n, {x, 0, n}], {n, 0, 17}]
A316186
Expansion of e.g.f. P(P(x)), where P(x) = Sum_{k>=1} prime(k)*x^k/k!.
Original entry on oeis.org
4, 18, 104, 687, 5064, 40934, 358083, 3346832, 33123000, 345219919, 3777134694, 43291666298, 518855171115, 6491738816768, 84656365477452, 1148895613585775, 16201725990730392, 237030534528945348, 3591398122456079285, 56254812062478841340, 909319044063443870702
Offset: 1
E.g.f.: A(x) = 4*x + 18*x^2/2! + 104*x^3/3! + 687*x^4/4! + 5064*x^5/5! + 40934*x^6/6! + ...
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p[x_] := p[x] = Sum[Prime[k] x^k/k!, {k, 21}]; a[n_] := a[n] = SeriesCoefficient[p[p[x]], {x, 0, n}]; Table[n! a[n], {n, 21}]
Showing 1-6 of 6 results.
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