A131182
Table T(n,k) = n!*k^n, read by upwards antidiagonals.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1
Offset: 0
The (inverted) table begins:
k=0: 1, 0, 0, 0, 0, 0, ... (A000007)
k=1: 1, 1, 2, 6, 24, 120, ... (A000142)
k=2: 1, 2, 8, 48, 384, 3840, ... (A000165)
k=3: 1, 3, 18, 162, 1944, 29160, ... (A032031)
k=4: 1, 4, 32, 384, 6144, 122880, ... (A047053)
k=5: 1, 5, 50, 750, 15000, 375000, ... (A052562)
k=6: 1, 6, 72, 1296, 31104, 933120, ... (A047058)
k=7: 1, 7, 98, 2058, 57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072, 98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).
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T:= (n,k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12); # Alois P. Heinz, Jan 06 2019
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from math import factorial
def A131182_T(n, k): # compute T(n, k)
return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022
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
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;
Sequences related to m values:
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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
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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
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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
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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
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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
A196258
a(n) = 11^n*n!.
Original entry on oeis.org
1, 11, 242, 7986, 351384, 19326120, 1275523920, 98215341840, 8642950081920, 855652058110080, 94121726392108800, 11388728893445164800, 1503312213934761753600, 214973646592670930764800, 33105941575271323337779200
Offset: 0
Cf.
A000142,
A000165,
A032031,
A047053,
A052562,
A047058,
A051188,
A051189,
A051232,
A051262,
A145448.
A052675
Expansion of e.g.f. (1-x)/(1-5*x).
Original entry on oeis.org
1, 4, 40, 600, 12000, 300000, 9000000, 315000000, 12600000000, 567000000000, 28350000000000, 1559250000000000, 93555000000000000, 6081075000000000000, 425675250000000000000, 31925643750000000000000
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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spec := [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
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Table[(4/5)*(5^n*n! + Boole[n==0]/4), {n, 0, 50}] (* G. C. Greubel, Jun 12 2022 *)
With[{nn=20},CoefficientList[Series[(1-x)/(1-5x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 31 2023 *)
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[4*factorial(n)*5^(n-1) + bool(n==0)/5 for n in (0..40)] # G. C. Greubel, Jun 12 2022
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