A340991 -id:A340991 - cp's OEIS frontend

A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 1, 10, 6, 1, 0, 5, 14, 21, 8, 1, 0, 1, 23, 47, 36, 10, 1, 0, 7, 28, 90, 108, 55, 12, 1, 0, 1, 49, 147, 258, 205, 78, 14, 1, 0, 1, 46, 249, 520, 595, 346, 105, 16, 1, 0, 1, 75, 360, 978, 1437, 1185, 539, 136, 18, 1

Offset: 0

Peter Luschny, Oct 03 2022

To clarify our terminology: We say T is the convolution triangle of a (or T is the partition transform of a), if a is a sequence of integers defined for n >= 1, and the transformation, as defined by the Maple function below, applied to a, returns T. In the generated lower triangular matrix T, i.e., in the convolution triangle of a, T(n, 1) = a(n) for n >= 1.
For instance, let a(n) = Bell(n), then we call A357583 the convolution triangle of the Bell numbers, but not A205574, which is also called the "Bell convolution triangle" in the comments but leads to a different triangle. Similarly, if a(n) = n!, then A090238 is the convolution triangle of the factorial numbers, not A084938. Third example: A128899 is the convolution triangle of the Catalan numbers in our setup, not A059365. More generally, we recommend that when computing the transform of a 0-based sequence to use only the terms for n >= 1 and not to shift the sequence.
Note that T is (0, 0)-based and the first column of a convolution triangle always is 1, 0, 0, 0, ... and the main diagonal is 1, 1, 1, 1, ... if a(1) = 1. The (1, 1)-based subtriangle of a genuine convolution triangle, i.e., a convolution triangle without column 0 and row 0, is often used in place of the convolution triangle but does not fit well into some applications of the convolution triangles.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,  1;
[3] 0, 3,  4,   1;
[4] 0, 1, 10,   6,   1;
[5] 0, 5, 14,  21,   8,   1;
[6] 0, 1, 23,  47,  36,  10,   1;
[7] 0, 7, 28,  90, 108,  55,  12,   1;
[8] 0, 1, 49, 147, 258, 205,  78,  14,  1;
[9] 0, 1, 46, 249, 520, 595, 346, 105, 16, 1;

Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, The P-transform.
Cormac O'Sullivan, De Moivre and Bell polynomials, arXiv:2203.02868 [math.CO], 2022.

Cf. A089026, A340991, A357588.

PMatrix := proc(dim, a) local n, k, m, g, M, A;
   if n = 0 then return [1] fi;
   A := [seq(a(i), i = 1..dim-1)];
   M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
   for m from 2 to dim do
       M[m, m] := M[m - 1, m - 1] * A[1];
       for k from m-1 by -1 to 2 do
           M[m, k] := add(A[i]*M[m-i, k-1], i = 1..m-k+1)
od od; M end:
a := n -> if isprime(n) then n else 1 fi: PMatrix(10, a);
# Alternatively, as the coefficients of row polynomials:
P := proc(n, x, a) option remember; ifelse(n = 0, 1,
    x*add(a(n - k)*P(k, x, a), k = 0..n-1)) end:
Pcoeffs := proc(n, a) seq(coeff(P(n, x, a), x, k), k=0..n) end:
seq(Pcoeffs(n, a), n = 0..9);
# Alternatively, term by term:
T := proc(n, k, a) option remember; # Alois P. Heinz style
    `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, a(n)),
    (q->add(T(j, q, a)*T(n-j, k-q, a), j=0..n))(iquo(k, 2)))) end:
seq(seq(T(n, k, a), k=0..n), n=0..9);

PMatrix[dim_, a_] := Module[{n, k, m, g, M, A}, If[n == 0, Return[1]]; A = Array[a, dim-1]; M = Array[0&, {dim, dim}]; M[[1, 1]] = 1; For[m = 2, m <= dim, m++, M[[m, m]] = M[[m-1, m-1]]*A[[1]]; For[k = m-1, k >= 2, k--, M[[m, k]] = Sum[A[[i]]*M[[m-i, k-1]], {i, 1, m-k+1}]]]; M];
a[n_] :=  If[PrimeQ[n], n, 1];
nmax = 10;
PM = PMatrix[nmax+1, a];
T[n_, k_] := PM[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 21 2022 *)

def ConvTriangle(dim: int, a) -> list[list[int]]:
    if callable(a): # Cache the input sequence.
        A = [a(i) for i in range(1, dim)]
    else:
        A = a
    print("In:", A)
    C = [[0 for k in range(m + 1)] for m in range(dim)]
    C[0][0] = 1
    for m in range(1, dim):
        C[m][m] = C[m - 1][m - 1] * A[0]
        for k in range(m - 1, 0, -1):
            C[m][k] = sum(A[i] * C[m - i - 1][k - 1] for i in range(m - k + 1))
    return C
from sympy import isprime, flatten
def a(n): return n if isprime(n) else 1
print(flatten(ConvTriangle(10, a)))

A014342 Convolution of primes with themselves.

Original entry on oeis.org

4, 12, 29, 58, 111, 188, 305, 462, 679, 968, 1337, 1806, 2391, 3104, 3953, 4978, 6175, 7568, 9185, 11030, 13143, 15516, 18177, 21150, 24471, 28152, 32197, 36678, 41543, 46828, 52621, 58874, 65659, 73000, 80949, 89462, 98631, 108396, 118869, 130102, 142071

Offset: 1

N. J. A. Sloane

			a(2)=12 because a(2) = prime(1)*prime(2) + prime(2)*prime(1) = 2*3 + 3*2 = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A023626, A024697, A025129, A209403.

Column k=2 of A340991.

a014342 n = a014342_list !! (n-1)
a014342_list= f (tail a000040_list) [head a000040_list] 1 where
   f (p:ps) qs k = sum (zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014, Mar 08 2012

[&+[NthPrime(n-i+1)*NthPrime(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016

A014342:=n->add(ithprime(i)*ithprime(n+1-i), i=1..n): seq(A014342(n), n=1..50); # Wesley Ivan Hurt, Dec 14 2016

Table[Sum[Prime[i] Prime[n + 1 - i], {i, n}], {n, 40}] (* Michael De Vlieger, Dec 13 2016 *)
Table[With[{p=Prime[Range[n]]},ListConvolve[p,p]],{n,40}]//Flatten (* Harvey P. Dale, May 03 2018 *)

{m=40;u=vector(m,x,prime(x));for(n=1,m,v=vecextract(u,concat("1..",n)); w=vector(n,x,u[n+1-x]);print1(v*w~,","))} \\ Klaus Brockhaus, Apr 28 2004

from numpy import convolve
from sympy import prime, primerange
def aupton(terms):
  p = list(primerange(2, prime(terms)+1))
  return list(convolve(p, p))[:terms]
print(aupton(41)) # Michael S. Branicky, Apr 12 2021

a(n) = Sum_{i=1..n} prime(i) * prime(n+1-i), where prime(i) is the i-th prime.

G.f.: (b(x)^2)/x, where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

More terms from Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001

A030018 Coefficients in 1/(1+P(x)), where P(x) is the generating function of the primes.

Original entry on oeis.org

1, -2, 1, -1, 2, -3, 7, -10, 13, -21, 26, -33, 53, -80, 127, -193, 254, -355, 527, -764, 1149, -1699, 2436, -3563, 5133, -7352, 10819, -15863, 23162, -33887, 48969, -70936, 103571, -150715, 219844, -320973, 466641, -679232, 988627, -1437185, 2094446, -3052743

Offset: 0

N. J. A. Sloane

sign

a(n+1)/a(n) => ~-1.456074948582689671... (see A072508). - Zak Seidov, Oct 01 2011

Seiichi Manyama, Table of n, a(n) for n = 0..6124 (terms 0..1000 from Zak Seidov)
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Backhouse's Constant

Cf. A000040, A072508, A340991.

a:= proc(n) option remember; `if`(n=0, 1,
      -add(ithprime(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 13 2018

max = 50; P[x_] := 1 + Sum[Prime[n]*x^n, {n, 1, max}]; s = Series[1/P[x], {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Sep 24 2014 *)

v=[];for(n=1,50,v=concat(v,-prime(n)-sum(i=1,n-1,prime(i)*v[#v-i+1])));v \\ Derek Orr, Apr 28 2015

Apply inverse of "INVERT" transform to primes: INVERT: a's from b's in 1+Sum a_i x^i = 1/(1-Sum b_i x^i).
a(n) = -prime(n) - Sum_{i=1..n-1} prime(i)*a(n-i), for n > 0. - Derek Orr, Apr 28 2015
a(n) = Sum_{k=0..n} (-1)^k * A340991(n,k). - Alois P. Heinz, Feb 01 2021

A030017 a(1) = 1, a(n+1) = Sum_{k = 1..n} p(k)*a(n+1-k), where p(k) is the k-th prime.

Original entry on oeis.org

1, 2, 7, 25, 88, 311, 1095, 3858, 13591, 47881, 168688, 594289, 2093693, 7376120, 25986209, 91549913, 322532092, 1136286727, 4003159847, 14103208628, 49685873471, 175044281583, 616684348614, 2172590743211, 7654078700221, 26965465508072, 94999850216565

Offset: 1

N. J. A. Sloane

Apply "INVERT" transform to primes.

			a(5) = 25*2 +7*3 +2*5 + 1*7 = 88.

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
N. J. A. Sloane, Transforms

Row sums of A340991(n-1).

Cf. A000040.

a:= proc(n) option remember; `if`(n=1, 1,
      add(a(n-i)*ithprime(i), i=1..n-1))
    end:
seq(a(n), n=1..29);  # Alois P. Heinz, Feb 10 2021

CoefficientList[ Series[ 1/(1 - Sum[ Prime[ n ]*x^n, {n, 1, 25} ] ), {x, 0, 25} ], x ]
(* Second program: *)
a[1] = 1; a[m_] := a[m] = Sum[Prime@ k  a[m - k], {k, m - 1}]; Table[a@ n, {n, 25}] (* Michael De Vlieger, Dec 13 2016 *)

INVERT: a's from b's in 1+Sum a_i x^i = 1/(1-Sum b_i x^i).

G.f: (1-b(x)/(b(x)-1))*x, where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

A030281 COMPOSE natural numbers with primes.

Original entry on oeis.org

2, 11, 53, 237, 1013, 4196, 16992, 67647, 265743, 1032827, 3979023, 15217248, 57835016, 218636365, 822691425, 3083074193, 11512489353, 42851360088, 159043175322, 588767623587, 2174488780469, 8013945343961, 29477541831841, 108233492257428, 396751988675780

Offset: 1

Christian G. Bower

Alois P. Heinz, Table of n, a(n) for n = 1..1823
N. J. A. Sloane, Transforms

Cf. A000040, A030017, A030282, A340991.

b:= proc(n) option remember; `if`(n=0, [1, 0], (p->
      p+[0, p[1]])(add(ithprime(j)*b(n-j), j=1..n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..27);  # Alois P. Heinz, Sep 11 2019

b[n_] := b[n] = If[n==0, {1, 0}, #+{0, #[[1]]}&[Sum[Prime[j] b[n-j], {j, 1, n}]]];
a[n_] := b[n][[2]];
Array[a, 27] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

G.f.: Sum_{j>=1} j*(Sum_{k>=1} prime(k)*x^k)^j. - Ilya Gutkovskiy, Apr 21 2019

a(n) = Sum_{k=0..n} k * A340991(n,k). - Alois P. Heinz, Feb 01 2021

A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.

Original entry on oeis.org

1, 3, 29, 291, 3121, 34123, 379853, 4280251, 48681569, 557686227, 6425630909, 74384480019, 864461820049, 10079577033243, 117859582680813, 1381492094548651, 16227770995740865, 190979248798795427, 2251327736286726749, 26579050506578504195, 314212180691846338801

Offset: 0

Alois P. Heinz, Feb 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..924

Cf. A000040, A309955, A340991.

b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, ithprime(n+1),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[k == 0, 1, If[k == 1, Prime[n + 1], With[{q = Quotient[k, 2]}, Sum[b[j, q] b[n - j, k - q], {j, 0, n}]]]];
a[n_] := b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Feb 04 2021, after Alois P. Heinz *)

a(n) = [x^(2n)] (Sum_{j>=1} prime(j)*x^j)^n.

a(n) = A340991(2n,n).

A014343 Three-fold convolution of primes with themselves.

Original entry on oeis.org

8, 36, 114, 291, 669, 1386, 2678, 4851, 8373, 13858, 22134, 34263, 51635, 75972, 109374, 154483, 214383, 292812, 394148, 523521, 686901, 891112, 1143936, 1454187, 1831973, 2288400, 2836044, 3488969, 4262541, 5173836, 6241612, 7486437, 8930649, 10598848

Offset: 0

N. J. A. Sloane

nonn

Cf. A000040, A014342.

Column k=3 of A340991.

Table[Sum[Prime[k + 1] Sum[Prime[i] Prime[# + 1 - i], {i, #}] &[n - k + 1], {k, 0, n}], {n, 0, 26}] (* Michael De Vlieger, Dec 13 2016 *)

From Mario C. Enriquez, Dec 13 2016: (Start)
G.f: (b(x)^3)/(x^2), where b(x) is the g.f. of A000040.
a(n) = Sum_{k=0..n} A014342(n-k+1)*A000040(k+1).
(End)

A014344 Four-fold convolution of primes with themselves.

Original entry on oeis.org

16, 96, 376, 1160, 3121, 7532, 16754, 34796, 68339, 127952, 229956, 398688, 669781, 1094076, 1742710, 2713604, 4139111, 6195712, 9115304, 13199072, 18833449, 26509260, 36843322, 50603884, 68740107, 92414192, 123039628, 162323200, 212312453, 275448380

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000040, A014342, A014343.

Column k=4 of A340991.

b:= proc(n, k) option remember; `if`(k=1, ithprime(n+1),
      add(b(j, floor(k/2))*b(n-j, ceil(k/2)), j=0..n))
    end:
a:= n-> b(n, 4):
seq(a(n), n=0..35);  # 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]], {j, 0, n}]];
a[n_] := b[n, 4];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

my(N = 50, x = 'x + O('x^N)); Vec(((1/x)*sum(k=1, N, prime(k)*x^k))^4) \\ Michel Marcus, Mar 10 2018

G.f.: ((1/x)*Sum_{k>=1} prime(k)*x^k)^4. - Ilya Gutkovskiy, Mar 10 2018

cp's OEIS Frontend

A357368 Triangle read by rows. Convolution triangle of the prime indicator sequence A089026.

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A014342 Convolution of primes with themselves.

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A030018 Coefficients in 1/(1+P(x)), where P(x) is the generating function of the primes.

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A030017 a(1) = 1, a(n+1) = Sum_{k = 1..n} p(k)*a(n+1-k), where p(k) is the k-th prime.

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A030281 COMPOSE natural numbers with primes.

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A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.

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Maple

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A014343 Three-fold convolution of primes with themselves.

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