A008543 -id:A008543 - cp's OEIS frontend

A049212 a(n) = -Product_{k=0..n} (10*k - 1); deca-factorial numbers.

1, 9, 171, 4959, 193401, 9476649, 559122291, 38579438079, 3047775608241, 271252029133449, 26853950884211451, 2927080646379048159, 348322596919106730921, 44933615002564768288809, 6245772485356502792144451, 930620100318118916029523199, 147968595950580907648694188641

Offset: 0

Wolfdieter Lang

Andrew Howroyd, Table of n, a(n) for n = 0..100
Index entries for sequences related to factorial numbers.

Cf. A008543, A035278, A048994, A049209, A049210, A049211, A048176.

[Round(10^n*Gamma(n+9/10)/Gamma(9/10)): n in [0..25]]; // G. C. Greubel, Feb 03 2022

CoefficientList[Series[(1-10*x)^(-9/10),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = {-prod(k=0, n, 10*k-1)} \\ Andrew Howroyd, Jan 02 2020

[10^n*rising_factorial(9/10, n) for n in (0..25)] # G. C. Greubel, Feb 03 2022

a(n) = 9*A035278(n) = (10*n-1)(!^10), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 10^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 10^n * Gamma(n+9/10) / Gamma(9/10). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-10*x)^(-9/10). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (20*k+9)*x - 10*(k+1)*(10*k+9)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-9*x-90*x^2/(1-29*x-380*x^2/(1-49*x-870*x^2/(1-69*x-1560*x^2/(1-89*x-2450*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(10*k+9)/(1 - x*(10*k+10)/Q(k+1)) (continued fraction).
G.f.: 1/(1-9*x/(1-10*x/(1-19*x/(1-20*x/(1-29*x/(1-30*x/(1-39*x/(1-40*x/(1-49*x/(1-50*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: Hypergeometric2F0([1, 9/10], --; 10*x). - G. C. Greubel, Feb 03 2022
Sum_{n>=0} 1/a(n) = 1 + (e/10)^(1/10)*(Gamma(9/10) - Gamma(9/10, 1/10)). - Amiram Eldar, Dec 22 2022

Terms a(14) and beyond from Andrew Howroyd, Jan 02 2020

A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

Original entry on oeis.org

1, 10, 210, 6720, 288960, 15603840, 1014249600, 77082969600, 6706218355200, 657209398809600, 71635824470246400, 8596298936429568000, 1126115160672273408000, 159908352815462823936000, 24465977980765812062208000, 4012420388845593178202112000

Offset: 0

Vaclav Kotesovec, Jan 28 2015

Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).

Cf. A000108, A254282, A254286, A254287.

m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!
FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]

m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by Georg Fischer, Dec 23 2019]
a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).
a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).
G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - Amiram Eldar, Dec 22 2022

A051151 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -6, 1, 72, -18, 1, -1296, 396, -36, 1, 31104, -10800, 1260, -60, 1, -933120, 355104, -48600, 3060, -90, 1, 33592320, -13716864, 2104704, -158760, 6300, -126, 1, -1410877440, 609700608, -102114432, 8772624, -423360, 11592, -168

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x-6*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -6,      1;
       72,    -18,      1;
    -1296,    396,    -36,    1;
    31104, -10800,   1260,  -60,   1;
  -933120, 355104, -48600, 3060, -90, 1;
   ...
3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3.

Wolfdieter Lang, First 10 rows.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are first introduced.]
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77. [Special cases of the numbers R_n^m(a,b) are tabulated.]

First (m=1) column sequence is: A047058(n-1).
Row sums (signed triangle): A008543(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A008542(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5).

a(n, m) = a(n-1, m-1) - 6*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: ((log(1 + 6*x)/6)^m)/m!.
a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

Various sections edited by Petros Hadjicostas, Jun 08 2020

A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

Original entry on oeis.org

1, 11, 253, 8855, 416185, 24554915, 1743398965, 144702114095, 13746700839025, 1470896989775675, 175036741783305325, 22929813173612997575, 3278963283826658653225, 508239308993132091249875, 84875964601853059238729125, 15192797663731697603732513375

Offset: 0

Nikolaos Pantelidis, Aug 06 2021

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), this sequence (m=12).

m:=12; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 16 2022

CoefficientList[Series[(1-12*x)^(-11/12),{x,0,20}], x] * Range[0, 20]!
FullSimplify[Table[12^n Gamma[n+11/12]/Gamma[11/12],{n,0,15}]] (* Stefano Spezia, Aug 07 2021 *)

m=12; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 16 2022

G.f.: 1/(1-11*x/(1-12*x/(1-23*x/(1-24*x/(1-35*x/(1-36*x/(1-47*x/(1-48*x/(1-59*x/(1-60*x/(1-...))))))))))) (Stieltjes continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(12*k+11)/(1 - x*(12*k+12)/Q(k+1) ) (continued fraction).
G.f.: 1/(1-11*x-132*x^2/(1-35*x-552*x^2/(1-59*x-1260*x^2/(1-83*x-2256*x^2/(1-107*x-3540*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/G(0) where G(k) = 1 - x*(24*k+11) - 12*(k+1)*(12*k+11)*x^2/G(k+1) (continued fraction).
a(n) = 12^n*Gamma(n+11/12)/Gamma(11/12). - Stefano Spezia, Aug 07 2021
Sum_{n>=0} 1/a(n) = 1 + (e/12)^(1/12)*(Gamma(11/12) - Gamma(11/12, 1/12)). - Amiram Eldar, Dec 22 2022

A144342 Lower triangular array called S2hat(-5) related to partition number array A144341.

Original entry on oeis.org

1, 5, 1, 55, 5, 1, 935, 80, 5, 1, 21505, 1210, 80, 5, 1, 623645, 29205, 1335, 80, 5, 1, 21827575, 782595, 30580, 1335, 80, 5, 1, 894930575, 27002800, 821095, 31205, 1335, 80, 5, 1, 42061737025, 1058476100, 27963925, 827970, 31205, 1335, 80, 5, 1, 2229272062325, 48782479625

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M32khat(-5)= A144341 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-5). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008543, A144344, A144345.

			[1];[5,1];[55,5,1];[935,80,5,1];[21505,1210,80,5,1];...

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Row sums A144343.

A144285 (S2hat(-4)).

a(n,m)=sum(product(|S2(-5;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S2(-5,n,1)|= A013988(n,1) = A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if n=1.

A144268 Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 55, 15, 1, 935, 220, 75, 30, 1, 21505, 4675, 2750, 550, 375, 50, 1, 623645, 129030, 70125, 30250, 14025, 16500, 1875, 1100, 1125, 75, 1, 21827575, 4365515, 2258025, 1799875, 451605, 490875, 211750, 144375, 32725, 57750, 13125, 1925, 2625, 105, 1, 894930575

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+4)-ary trees if the outdegree is r >= 0.
If M32(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle A013988(n,m)= |S2(-5;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=75. The relevant partition of 4 is (2^2). The 75 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are 5-ary because r=1 vertices are 5-ary and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 5^2=25 versions due to the two 5-ary root vertices.

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-5,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-5,j,1)|^e(n,k,j),j=1..n), with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A057130 Product of first n primes of form 6k-1.

Original entry on oeis.org

5, 55, 935, 21505, 623645, 25569445, 1201763915, 63693487495, 3757915762205, 266812019116555, 22145397586674065, 1970940385213991785, 199064978906613170285, 21299952743007609220495, 2406894659959859841915935, 315303200454741639290987485

Offset: 1

Henry Bottomley, Aug 11 2000

			a(3) = 5*11*17 = 935.
a(4) = 21505 = 5 * 11 * 17 * 23.

Cf. A000040, A002110, A002476, A006278, A007528, A074040, A078586, A121940.

Cf. A008543, A057131.

lista(nn) = {pp = 1; for (n = 1, nn, p = prime(n); if (Mod(p, 6) == -1, pp *= p; print1(pp, ", ")););} \\ Michel Marcus, Sep 08 2013

a(n) = a(n-1)*A007528(n) = (A057131(n)+1)/6.

More terms from Larry Reeves (larryr(AT)acm.org), Oct 06 2000

More terms from Michel Marcus, Sep 08 2013

A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 7, 6, 0, 6, 29, 46, 24, 0, 24, 146, 329, 326, 120, 0, 120, 874, 2521, 3604, 2556, 720, 0, 720, 6084, 21244, 39271, 40564, 22212, 5040, 0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320

Offset: 0

Philippe Deléham, Dec 01 2003, Aug 17 2007

			Triangle begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     7,      6;
  0,    6,    29,     46,     24;
  0,   24,   146,    329,    326,    120;
  0,  120,   874,   2521,   3604,   2556,    720;
  0,  720,  6084,  21244,  39271,  40564,  22212,   5040;
  0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

Variant: A059364, diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A084938.

A088996:= func< n,k | (&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n,n-j): j in [0..n]]) >;
[A088996(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022

A059364 := (n, k) -> add(abs(Stirling1(n, n - j))*binomial(j, n - k), j = 0..n);
seq(seq(A059364(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Aug 27 2025

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n,0,10}, {k,n+1,0,-1}] (* Michael De Vlieger, Jun 19 2018 *)

def A088996(n,k): return add((-1)^(n-i)*binomial(i,k)*stirling_number1(n+1,n+1-i) for i in (0..n))
for n in (0..10): [A088996(n,k) for k in (0..n)]  # Peter Luschny, May 12 2013

T(n, k) given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. [Original name.]
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022

New name using a formula of G. C. Greubel by Peter Luschny, Aug 27 2025

A144341 Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 55, 5, 1, 935, 55, 25, 5, 1, 21505, 935, 275, 55, 25, 5, 1, 623645, 21505, 4675, 3025, 935, 275, 125, 55, 25, 5, 1, 21827575, 623645, 107525, 51425, 21505, 4675, 3025, 1375, 935, 275, 125, 55, 25, 5, 1, 894930575, 21827575, 3118225, 1182775, 874225, 623645

Offset: 1

Wolfdieter Lang Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
If M32hat(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-5):= A144342(n,m).

			a(4,3)= 25 = |S2(-5,2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A144284 (M32hat(-4) array).

a(n,k)= product(|S2(-5,j,1)|^e(n,k,j),j=1..n) with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

Formally a(n,k)= 'M32(-5)/M3' = 'A144268/A036040' (elementwise division of arrays).

A153271 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.

Original entry on oeis.org

5, 5, 30, 5, 35, 315, 5, 40, 440, 6160, 5, 45, 585, 9945, 208845, 5, 50, 750, 15000, 375000, 11250000, 5, 55, 935, 21505, 623645, 21827575, 894930575, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 5, 65, 1365, 39585, 1464645, 65909025, 3493178325, 213083877825, 14702787569925

Offset: 0

Roger L. Bagula, Dec 22 2008

Row sums are {5, 35, 355, 6645, 219425, 11640805, 917404295, 101177741765, 14919432040765, 2839006665525525, 677815000136926955, ...}.

			Triangle begins as:
  5;
  5, 30;
  5, 35, 315;
  5, 40, 440,  6160;
  5, 45, 585,  9945, 208845;
  5, 50, 750, 15000, 375000, 11250000;
  5, 55, 935, 21505, 623645, 21827575, 894930575;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).
Sequences related to m values:
m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.
m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.

m:=3;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019

m:=3; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019

T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten

T(n,k) = my(m=3); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019

def T(n, k):
    m=3
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3.

Edited by G. C. Greubel, Dec 03 2019

cp's OEIS Frontend

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