A093883 -id:A093883 - cp's OEIS frontend

A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A006963, A076756, A093883, A296590.

[(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j; z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]           (* A093883 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)
Table[v[n]/d[n], {n,20}]     (* A203469 *)
(* Second program *)
Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

[product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

a(n) = Product_{i=1..n} binomial(2n-i,i).  - Enrique Pérez Herrero, Feb 20 2013
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^n/sqrt(Pi))^n*BarnesG(n+3/2)/(BarnesG(n+2)*BarnesG(3/2)).
a(n) = (n!/2^(n-1))*Product_{j=1..n-1} Catalan(j). (End)
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^(n/2 + 1/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 26 2023

A218566 Triangle T[r,c]=(r-1)binomial(r-1,c-1)(c-1)!*A093883(c), read by rows.

Original entry on oeis.org

0, 1, 3, 2, 12, 240, 3, 27, 1080, 226800, 4, 48, 2880, 1209600, 3657830400, 5, 75, 6000, 3780000, 22861440000, 1267438233600000, 6, 108, 10800, 9072000, 82301184000, 9125555281920000, 11274806061917798400000

Offset: 1

M. F. Hasler, Nov 02 2012

T[b,d] gives the number of positive numbers that can be written in base b with d(d+1)/2 digits such that for each k=1,...,d some digit appears exactly k times, cf. A218560, A167819, A218556 and related sequences.

			The first 6 rows of the triangle are:
r=1: 0;
r=2: 1, 3;
r=3: 2, 12,  240;
r=4: 3, 27,  1080,  226800;
r=5: 4, 48,  2880,  1209600,  3657830400;
r=6: 5, 75,  6000,  3780000,  22861440000,  1267438233600000.
Row 2 counts the numbers 1 and 4=100[2], 5=101[2], 6=110[2].
Row 3 counts the numbers {1, 2} and {9=100[3], 10=101[3], 12=110[3], 14=112[3], 16=121[3], ..., 25=221[3]} and {248=100012[3], ..., 714=222110[3]}.

T(r,c)=(r-1)*binomial(r-1,c-1)*(c-1)!*A093883(c)

T[r,1] = r-1. T[r,2] = 3(r-1)^2. T[r,3] = 60(r-2)(r-1)^2, etc.

A006963 Number of planar embedded labeled trees with n nodes: (2*n-3)!/(n-1)! for n >= 2, a(1) = 1.

Original entry on oeis.org

1, 1, 3, 20, 210, 3024, 55440, 1235520, 32432400, 980179200, 33522128640, 1279935820800, 53970627110400, 2490952020480000, 124903451312640000, 6761440164390912000, 393008709555221760000, 24412776311194951680000, 1613955767240110694400000

Offset: 1

Simon Plouffe and N. J. A. Sloane

For n>1: central terms of the triangle in A173333; cf. A001761, A001813. - Reinhard Zumkeller, Feb 19 2010
Can be obtained from the Vandermonde permanent of the first n positive integers; see A093883. - Clark Kimberling, Jan 02 2012
All trees can be embedded in the plane, but "planar embedded" means that orientation matters but rotation doesn't. For example, the n-star with n-1 edges has n! ways to label it, but rotation removes a factor of n-1. Another example, the n-path has n! ways to label it, but rotation removes a factor of 2. - Michael Somos, Aug 19 2014

			G.f. = x + x^2 + 3*x^3 + 20*x^4 + 210*x^5 + 3024*x^6 + 55440*x^7 + 1235520*x^8 + ...
a(5) = 210 = 30 + 60 + 120 where 30 is for the star, 60 for the path, and 120 for the tree with one trivalent vertex. - _Michael Somos_, Aug 19 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
David Callan, A quick count of plane (or planar embedded) labeled trees.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, Journal of Knot Theory and Its Ramifications, Vol. 25, No. 8 (2016), 1650047; arXiv preprint, arXiv:1507.03163 [math.CO], 2015-2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 109.
Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80. (Annotated scanned copy)
J. W. Moon, Counting Labelled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.
Ran J. Tessler, A Cayley-type identity for trees, arXiv:1809.00001 [math.CO], 2018.
Index entries for sequences related to trees.

Cf. A000407, A001761, A001813, A173333.

[1] cat [Factorial(2*n-3)/Factorial(n-1): n in [2..20]]; // Vincenzo Librandi, Nov 12 2011

1, seq((2*n-3)!/(n-1)!,n=2..30); # Robert Israel, Aug 20 2014

Join[{1},Table[(2n-3)!/(n-1)!,{n,2,20}]] (* Harvey P. Dale, Nov 03 2011 *)
a[ n_] := With[{m = n - 1}, If[m < 1, Boole[m == 0], m! SeriesCoefficient[ -Log[(1 + Sqrt[1 - 4 x]) / 2], {x, 0, m}]]] (* Michael Somos, Jul 01 2013 *)
a[ n_] := If[n < 2, Boole[n == 1], (2 n - 3)! / (n - 1)!]; (* Michael Somos, Aug 19 2014 *)
a[1] := 1; a[n_] := (-1)^(n - 1)*Sum[(-1)^k*Binomial[2*n - 3, n + k - 2]*StirlingS1[n + k - 1, k + 1], {k, 1, n - 1}]; Flatten[Table[a[n], {n, 1, 19}]] (* Detlef Meya, Jan 18 2024 *)

{a(n) = n--; if( n<1, n==0, n! * polcoeff( -log( (1 + sqrt(1 - 4*x + x * O(x^n))) / 2), n))}; /* Michael Somos, Jul 01 2013 */

def A006963(n): return 1 if n==1 else factorial(2*n-3)/factorial(n-1)
[A006963(n) for n in range(1,31)] # G. C. Greubel, May 23 2023

E.g.f. for a(n+1), n >= 1, log(c(x)); c(x) = g.f. for Catalan numbers A000108. - Wolfdieter Lang
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n * erfc(sqrt(x)/2)/2, x=0..infinity), n=0, 1..., where erfc(x) is the complementary error function. - Karol A. Penson, Sep 27 2001
a(n) ~ 2^(-5/2)*n^-2*2^(2*n)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
a(n+1) = (n+1)*(n+2)*...*(2n-1) for n>=2. - Jaroslav Krizek, Nov 09 2010
E.g.f. (A(x)-1) is reversion of exp(-x)-exp(-2*x). - Vladimir Kruchinin, Jan 30 2012
G.f.: 1 + x*G(0) where G(k) = 1 + x*(2*k+1)*(4*k+3)/(k + 1 - 4*x*(k+1)^2*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
E.g.f.: 1 + x*E(0) where E(k) = 1 + x*(2*k+1)*(4*k+3)/(2*(k + 1)^2 - 8*x*(k+1)^3*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
E.g.f: sqrt(1-4*x)/4 - 1/4 + 3*x/2 - x*log((1+sqrt(1-4*x))/2). - Robert Israel, Aug 20 2014
D-finite with recurrence (-n+1)*a(n) +2*(2*n-3)*(n-2)*a(n-1)=0. - R. J. Mathar, Jan 03 2018
From Amiram Eldar, Apr 03 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/2 + 3*exp(1/4)*sqrt(Pi)*erf(1/2)/4, where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - sqrt(Pi)*erfi(1/2)/(4*exp(1/4)), where erfi is the imaginary error function. (End)
a(n) = A000407(n-2)/(n-1). - R. J. Mathar, Mar 30 2023
a(1) = 1; a(n) = (-1)^(n - 1)*Sum_{k=1..n - 1} (-1)^k*binomial(2*n - 3, n + k - 2)*Stirling1(n + k - 1, k + 1). - Detlef Meya, Jan 18 2024

A203312 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 3, 147, 298116, 47460365316, 965460013501733568, 3717096745012192786213464768, 3763515081241454304168766426610670649344, 1329626784930718063722475681347135527472012731205697536

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A203012, A203673, A367543.

f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203312 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)

from operator import mul
from functools import reduce
def v(n): return 1 if n==1 else reduce(mul, [j**2 - j*k + k**2 for k in range(2, n + 1) for j in range(1, k)])
print([v(n) for n in range(1, 11)]) # Indranil Ghosh, Jul 26 2017

a(n) ~ c * n^(n^2 - n - 2/3) / exp(3*n^2/2 - n*(n+1)*Pi/(2*sqrt(3)) - n), where c = Gamma(1/3) * 3^(1/12) * exp(Pi/(12*sqrt(3))) / (2^(4/3) * Pi^(4/3)) = 0.2945280196744096322469352538791946777977998766871923997662057483092872... - Vaclav Kotesovec, Nov 22 2023

A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

Original entry on oeis.org

1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203476.

G. C. Greubel, Table of n, a(n) for n = 1..28

Cf. A000290, A093883, A202768, A203476, A272244, A293290, A323755, A324403, A367517.

[(&*[(&*[j^2 + k^2: k in [1..j]])/(2*j^2): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

a:= n-> mul(mul(i^2+j^2, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)

[product(product(j^2+k^2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4), where c = A323755 = sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8) = 0.274528350333552903800408993482507428142383783773190451181... - Vaclav Kotesovec, Jan 26 2019

Name edited by Alois P. Heinz, Jul 23 2017

A203673 Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 21, 254163, 11213968422384, 6451450005117349260375984, 127857993263632065817610313129228311433216, 191199773886534869435599958788731398661833328276349525268803584

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A367550.

f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203673 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203674 *)

From Vaclav Kotesovec, Nov 22 2023: (Start)
a(n) = A203012(n) * A203312(n).
a(n) ~ c * 3^(n*(3*n+1)/4) * n^(2*n^2 - 2*n - 3/2) / exp(3*n^2 - n*(n+1)*Pi*sqrt(3)/4 - 2*n), where c = Gamma(1/3)^(3/2) * 3^(7/24) * exp(Pi/(8*sqrt(3))) / (2^(5/2) * Pi^(5/2)) = 0.076580853261060033865281896312127877504662138809362419847380161198324... (End)

A080358 Value of Vandermonde determinant for the first n prime numbers: V[prime(1), ..., prime(n)].

Original entry on oeis.org

1, 1, 1, 6, 240, 414720, 4379443200, 2648687247360000, 11619303595714805760000, 4047756373260469165621248000000, 311107430628520522709128328175943680000000, 152539657943794787580793302587123569672794931200000000

Offset: 0

Labos Elemer, Feb 19 2003

nonn

The value of the Vandermonde determinant is unchanged if the numbers are shifted by an arbitrary constant c, i.e., V[prime(1), ..., prime(n)] = V[prime(1)-c, ..., prime(n)-c].

For a guide to related sequences, see A093883. - Clark Kimberling, Jan 03 2012

			a(1)=1 corresponds to 1 X 1 V-matrix, while a(2)=1 is computed from a 2 X 2 matrix.
n = 2: a(2) = prime(2) - prime(1) = 3 - 2 = 1;
n = 3: a(3) = (5-3)*(5-2)*(3-2) = 2*3*1 = 6; ...
n = 6: a(6) = (13-11)*(13-7)*(13-5)*(13-3)*(13-2)*(11-7)*(11-5)*(11-3)*(11-2)*(7-5)*(7-3)*(7-2)*(5-3)*(5-2)*(3-2) = 2*6*8*10*11*4*6*8*9*2*4*5*2*3*1 = 10560*1728*40*6*1 = 4379443200.

Alois P. Heinz, Table of n, a(n) for n = 0..36

Cf. A000040, A001359, A203521, A290179.

with(LinearAlgebra):
a:= n-> Determinant(Matrix(n, (i,j)-> ithprime(i)^(j-1))):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 22 2017

b[x_] := Prime[x] d[x_] := b[x+1]-b[x] t[m_] := b[m+1]-Table[b[x], {x, 1, m}] pt[x_] := Apply[Times, t[x]] va[x_] := Apply[Times, Table[pt[w], {w, 1, x}]] Table[va[j], {j, 1, 10}]

a(n) = Product_{i, j, i>j} (prime(i) - prime(j)). a(n) is the product of binomial(n, 2) prime differences of not necessarily consecutive primes.

For n > 1, a(n) = sqrt(Delta_n), where Delta_n is the discriminant of the polynomial (x - 2)*(x - 3)*...*(x - prime(n)). - Thomas Ordowski, Mar 15 2023

A203012 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 7, 1729, 37616124, 135933424914924, 132432199651531695045312, 51437933151214684812682944045953088, 11056394929890243558409721156996503083526683082752, 1743892714865607005898689849291524734866677095031979100765833773056

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

			a(1)=1
a(2)=1^2+1*2+2^2=7
a(3)=(1^2+1*2+2^2)(1^3+1*3+3^2)(2^2+2*3+3^2)=1729.

Cf. A203312, A203475, A203673, A367542.

f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 + f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203012 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203158 *)

a(n) ~ c * n^(n^2 - n - 5/6) * 3^(n*(3*n+1)/4) / exp(3*n^2/2 - n - n*(n+1)*Pi / (4*sqrt(3))), where c = sqrt(Gamma(1/3)) * 3^(5/24) * exp(Pi/(24*sqrt(3))) / (2^(7/6) * Pi^(7/6)) = 0.26001211479205772659823692637002123572622409280442625312217301129630097... - Vaclav Kotesovec, Nov 22 2023

A203311 Vandermonde determinant of (1,2,3,...,F(n+1)), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 2, 48, 30240, 1596672000, 18172937502720000, 122457316443772566896640000, 1284319496829094129116119090331648000000, 55603466527142141932748234118927499493985767915520000000, 26110840958525805673462196263372614726154694067746586937781385166848000000000

Offset: 0

Clark Kimberling, Jan 01 2012

nonn

Each term divides its successor, as in A123741. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203313.

For a signed version, see A123742. For a guide to related sequences, including sequences of Vandermonde permanents, see A093883.

			v(4) = (2-1)*(3-1)*(3-2)*(5-1)*(5-2)*(5-3).

Cf. A000045, A123741, A123742, A203313, A203518.

with(LinearAlgebra): F:= combinat[fibonacci]:
a:= n-> Determinant(VandermondeMatrix([F(i)$i=2..n+1])):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]
Table[v[n], {n, 1, z}]                (* A203311 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A123741 *)
Table[v[n]/d[n], {n, 1, 13}]          (* A203313 *)

from sympy import fibonacci, factorial
from operator import mul
from functools import reduce
def f(j): return fibonacci(j + 1)
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [f(k) - f(j) for j in range(1, k)]) for k in range(2, n + 1)])
print([v(n) for n in range(1, 16)]) # Indranil Ghosh, Jul 26 2017

a(n) ~ c * d^n * phi^(n^3/3 + n^2/2) / 5^(n^2/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio, d = 0.120965069090607877853843907542896935455225485213927649233956250456604334... and c = 197.96410442333389877538426269... - Vaclav Kotesovec, Apr 08 2021

A203482 a(n) = Product_{1 <= i < j <= n} (i! + j!).

Original entry on oeis.org

1, 3, 168, 3276000, 877449500928000, 207244701437748852512194560000, 4000516840149319128119305958853265913416777728000000, 796608816253064941944831363792070377592412324940256242675178274726476775424000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203483.

See A093883 for a guide to related sequences.

G. C. Greubel, Table of n, a(n) for n = 1..16

Cf. A000142, A203483, A203510, A306729, A325052.

[(&*[(&*[Factorial(j) + Factorial(k): k in [1..j]])/(2*Factorial(j)): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

a:= n-> mul(mul(i!+j!, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= j!; z = 10;
v[n_]:= Product[Product[f[k] + f[j], {j, k-1}], {k, 2, n}]
d[n_]:= Product[(i-1)!, {i, n}]  (* A000178 *)
Table[v[n], {n, z}]              (* A203482 *)
Table[v[n+1]/v[n], {n, z-1}]     (* A203483 *)
Table[v[n]/d[n], {n, 10}]        (* A203510 *)
(* Second program *)
Table[Product[j!+k!, {j,n}, {k,j-1}], {n,15}] (* G. C. Greubel, Aug 29 2023 *)

[product(product(factorial(j) + factorial(k) for k in range(1,j)) for j in range(1,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 29 2023

a(n) ~ c * n^(n^3/3 + n^2/4 - 7*n/12 + 5/8) * (2*Pi)^(n*(n-1)/4) / exp(4*n^3/9 - n^2/8 - n), where c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... - Vaclav Kotesovec, Nov 20 2023

Name edited by Alois P. Heinz, Jul 23 2017

cp's OEIS Frontend

A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

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A218566 Triangle T[r,c]=(r-1)*binomial(r-1,c-1)*(c-1)!*A093883(c), read by rows.

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A006963 Number of planar embedded labeled trees with n nodes: (2*n-3)!/(n-1)! for n >= 2, a(1) = 1.

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A203312 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).

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A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

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SageMath

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A203673 Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

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Mathematica

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A080358 Value of Vandermonde determinant for the first n prime numbers: V[prime(1), ..., prime(n)].

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Maple

A218566 Triangle T[r,c]=(r-1)binomial(r-1,c-1)(c-1)!*A093883(c), read by rows.