A014345
Exponential convolution of primes with themselves.
Original entry on oeis.org
4, 12, 38, 118, 362, 1082, 3166, 8910, 24426, 64226, 165262, 413418, 1021362, 2490686, 6009150, 14401410, 34098042, 80281962, 187356750, 432549154, 992941250, 2256712462, 5088826238, 11408805862, 25425739346, 56383362854, 124565557898, 274390550594
Offset: 0
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[&+[NthPrime(k+1)*NthPrime(n-k+1)*Binomial(n, k): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Jun 07 2019
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a:= proc(n) option remember; (p-> add(
p(j+1)*p(n-j+1)*binomial(n, j), j=0..n))(ithprime)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 10 2018
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a[n_] := Sum[Prime[j + 1] Prime[n - j + 1] Binomial[n, j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 05 2018, from Maple *)
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{a(n) = sum(j=0,n, binomial(n,j)*prime(j+1)*prime(n-j+1))}; \\ G. C. Greubel, Jun 07 2019
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[sum(binomial(n,j)*nth_prime(j+1)*nth_prime(n-j+1) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 07 2019
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
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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
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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 *)
A014351
Four-fold exponential convolution of primes with themselves (divided by 8).
Original entry on oeis.org
2, 12, 74, 460, 2861, 17722, 109037, 665020, 4014521, 23954342, 141123193, 820074040, 4697137637, 26504081542, 147300078809, 806343223508, 4349380581953, 23130233881414, 121379963732665, 629130600591920, 3224186845616653, 16354295398317790, 82187373706636505
Offset: 0
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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)/8:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 07 2018
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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]/8;
a /@ Range[0, 30] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)
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), ...
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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
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Table[n! SeriesCoefficient[Sum[Prime[k + 1] x^k/k!, {k, 0, n}]^n, {x, 0, n}], {n, 0, 17}]
Showing 1-4 of 4 results.