A000142 -id:A000142 - cp's OEIS frontend

A067109 Number of occurrences of the string n in n! (A000142).

1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 2, 4, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 2, 5, 0

Offset: 1

Amarnath Murthy, Jan 08 2002

a(A033180(n)) > 0. - Reinhard Zumkeller, Aug 23 2008

			a(4) = 1 as 4! = 24 and 4 occurs once;
a(5) = 0 as 5! = 120 does not contain a 5;
a(20) = 1 as 20! = 2432902008176640000 and 20 occurs once.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000142, A033180.

import Data.List (tails, isPrefixOf)
a067109 n = sum $
   map (fromEnum . (show n `isPrefixOf`)) (tails $ show $ a000142 n)
-- Reinhard Zumkeller, Aug 28 2014

Table[ Length[ StringPosition[ ToString[n! ], ToString[n]]], {n, 1, 75} ]
Table[SequenceCount[IntegerDigits[n!],IntegerDigits[n],Overlaps->True],{n,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2019 *)

More terms from Robert G. Wilson v, Jan 09 2002

A082765 Trinomial transform of the factorial numbers (A000142).

Original entry on oeis.org

1, 4, 45, 1282, 70177, 6239016, 817234189, 147950506390, 35370826189857, 10791515504716012, 4091225768720823181, 1886585105032464025674, 1039774852573506696192385, 674970732343624159361034832

Offset: 0

Emanuele Munarini, May 21 2003

Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a sequence of n (possibly empty) lists, each of length at most 2. - Bob Proctor, Apr 18 2005

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05, 2007
Index entries for related partition-counting sequences

a(n) = Sum[C(n, k)*A099022(k), 0<=k<=n]
Replace "sequence" by "collection" in comment: A105747.
Replace "lists" by "sets" in comment: A003011.

a(n) = Sum[ Trinomial[n, k] k!, {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907)

Integral_{x=0..infinity} (x^2+x+1)^n*exp(-x) dx - Gerald McGarvey, Oct 14 2006

A089833 a(n) = A000108(n)*(A000142(n+1)-1).

Original entry on oeis.org

0, 1, 10, 115, 1666, 30198, 665148, 17296851, 518916970, 17643220738, 670442556004, 28158587998814, 1295295050441588, 64764752531737100, 3497296636751245560, 202843204931717665155, 12576278705767060962330

Offset: 0

Antti Karttunen, Dec 05 2003

nonn

The first column of A089831.

Cf. A000108, A001813, A089835.

a(n) = A001813(n) - A000108(n).

A102409 Even triangle n!. This table read by rows gives the coefficients of sum formulas of n-th factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+3, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, 0, 0, 0, -20, 8, 0, 0, 20280, -6530, -1275, 362, 3, 0, -8749440, 21627600, -4871940, -66510, 48300, 390, 0, -261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0, -974260634054400, -1140185248443360, 353509119454680, -8136128999880, -3234018579750

Offset: 1

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101751.

			Triangle starts:
0, 1, 0, 0;
0, -20, 8, 0, 0;
20280, -6530, -1275, 362, 3, 0;
-8749440, 21627600, -4871940, -66510, 48300, 390, 0;
-261763004160, 72965762016, 13117344800, -3757930680, 72406040, 13101144, 90440, 0;
...
11!=39916800; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing 11!.
=> 11! = [ -974260634054400 -1140185248443360*11 +353509119454680*11^2 -8136128999880*11^3 -3234018579750*11^4 +109743298560*11^5 +6053880420*11^6 +34067880*11^7 +9450*11^8 ]/10! = 39916800.

Cf. A102410, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.

A102410 Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

1, 0, 0, -6, 3, 1, 0, 2400, -2024, 264, 32, 0, 2570400, 909720, -666540, 55800, 3420, 0, -19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840, -219303218534400, -11953192930560, 27128332828800, -2808016545600, -125442525600, 14164990560, 280576800

Offset: 1

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

			Triangle starts:
1, 0, 0;
-6, 3, 1, 0;
2400, -2024, 264, 32, 0;
2570400, 909720, -666540, 55800, 3420, 0;
-19071521280, 12195884736, -762499920, -282106440, 22425480, 741384, 840;
...
11!=39916800; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing 11!.
=> 11! = [ -219303218534400 -11953192930560*11 +27128332828800*11^2 -2808016545600*11^3 -125442525600*11^4 +14164990560*11^5 +280576800*11^6 +453600*11^7 ]/10! = 39916800.

Cf. A102409, A008276, A094216, A000142, A094638, A101751, A102411, A102412, A101752, A003422, A101559, A101032, A099731.

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset: 0

Alford Arnold, Aug 19 2006

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

			Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Vincenzo Librandi, Rows n = 0..100, flattened
J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
Cf. A008279.
Cf. A008277, A132159 (mirrored).

a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* Harvey P. Dale, Apr 25 2011 *)

A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330.

Original entry on oeis.org

1, 2, 6, 2, 24, 24, 120, 240, 24, 720, 2400, 720, 5040, 25200, 15120, 720, 40320, 282240, 282240, 40320, 362880, 3386880, 5080320, 1451520, 40320, 3628800, 43545600, 91445760, 43545600, 3628800, 39916800, 598752000, 1676505600, 1197504000, 199584000, 3628800

Offset: 0

Tom Copeland, Oct 30 2007, Nov 29 2007, Nov 30 2007

Construct the infinite array of polynomials
a(0,t) = 1
a(1,t) = 2
a(2,t) = 6 + 2 t
a(3,t) = 24 + 24 t
a(4,t) = 120 + 240 t + 24 t^2
a(5,t) = 720 + 2400 t + 720 t^2
a(6,t) = 5040 + 25200 t + 15120 t^2 + 720 t^3
This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.
b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2.
Then A000165(n) = a(n,1).
Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2).
n! * A000129(n+1) = a(n,2) = A110327(n,0).
A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,2).
A000142(n+1) = a(n,0).
From Peter Bala, Sep 09 2013: (Start)
Let {P(n,x)}n>=0 be a polynomial sequence. Koutras has defined generalized Eulerian numbers associated with the sequence P(n,x) as the coefficients A(n,k) in the expansion of P(n,x) in a series of factorials of degree n, namely P(n,x) = Sum_{k=0..n} A(n,k)* binomial(x+n-k,n). The choice P(n,x) = x^n produces the classical Eulerian numbers of A008292. Let now P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial polynomial. Then the present table is the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x). See A228955 for the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x + 1). (End)

			Triangle begins as:
        1;
        2;
        6,        2;
       24,       24;
      120,      240,       24;
      720,     2400,      720;
     5040,    25200,    15120,      720;
    40320,   282240,   282240,    40320;
   362880,  3386880,  5080320,  1451520,   40320;
  3628800, 43545600, 91445760, 43545600, 3628800;

M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.

Cf. A228955.

Flat(List([0..10], n-> List([0..Int(n/2)], k-> Factorial(n)*Binomial(n+1, 2*k+1) ))); # G. C. Greubel, Dec 30 2019

[Factorial(n)*Binomial(n+1, 2*k+1): k in [0..Floor(n/2)], n in [0..10]]; // G. C. Greubel, Dec 30 2019

for n from 0 to 10 do
seq( n!*binomial(n+1,2*k+1), k = 0..floor(n/2) )
end do; # Peter Bala, Sep 09 2013

Table[n!*Binomial[n+1, 2*k+1], {n,0,10}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Dec 30 2019 *)

T(n,k) = n!*binomial(n+1, 2*k+1);
for(n=0,10, for(k=0, n\2, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 30 2019

[[factorial(n)*binomial(n+1, 2*k+1) for k in (0..floor(n/2))] for n in (0..10)] # G. C. Greubel, Dec 30 2019

E.g.f. for the polynomials b(.,t), introduced above, is 1 - 2x - (t-1) * x^2; therefore e.g.f. for the polynomials a(.,t), which are the row polynomials of this array, is 1 / ( 1 - 2x - (t-1) * x^2 ) = (t-1) / ( t - ( 1 + x*(t-1) )^2 ).
Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. at u = 1/t. Compare A076743.
a(n,t) = n!*Sum_{k>=0} binomial(n+1,2k+1) * t^k = n!*Sum_{k>=0} A034867(n,k) * t^k.
Additional relations are given by formulas in A133314.
From Peter Bala, Sep 09 2013: (Start)
Recurrence equation: T(n+1,k) = (n+2 +2*k)T(n,k) + (n +2 -2*k)T(n,k-1).
Let P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial.
T(n,k) = Sum_{j=0..k+1} (-1)^(k+1-j)*binomial(n+1,k+1-j)*P(n,2*j) for n >= 1.
The row polynomial a(n,t) satisfies t*a(n,t)/(1 - t)^(n+1) = Sum_{j>=1} P(n,2*j)*t^j. For example, for n = 3 we have t*(24 + 24*t)/(1 - t)^4 = 2*3*4*t + (4*5*6)*t^2 + (6*7*8)*t^3 + ..., while for n = 4 we have t*(120 + 240*t + 24*t^2)/(1 - t)^5 = (2*3*4*5)*t + (4*5*6*7)*t^2 + (6*7*8*9)*t^3 + .... (End)

Removed erroneous and duplicate statements. - Tom Copeland, Dec 03 2013

A280062 a(n) = A049502(A000142(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 13, 16, 22, 38, 47, 73, 60, 127, 160, 166, 194, 249, 348, 345, 359, 497, 532, 682, 709, 727, 1000, 887, 1312, 1155, 1297, 1934, 2069, 1722, 1796, 2148, 2337, 1839, 2595, 2774, 2440, 3314, 3450, 3253, 3379, 3786, 4466, 4366, 4795, 5189, 5598

Offset: 0

Indranil Ghosh, Jan 04 2017

nonn

a(n) is the major index (2nd definition) of n!.

			for n=15, A000142(n) = 1307674368000 and A049502(1307674368000) = 166. So a(n) = 166.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A049502, A000142.

import math
def m(N):
    x=bin(int(N))[2:][::-1]
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s
a=lambda n: m(math.factorial(n))

A373433 a(n) = A000111(n) * A000142(n). Row sums of A373434.

Original entry on oeis.org

1, 1, 2, 12, 120, 1920, 43920, 1370880, 55843200, 2879815680, 183330604800, 14122244505600, 1294628759424000, 139287595371724800, 17379949655535667200, 2489494639794978816000, 405724534220435189760000, 74646464089618378653696000, 15396938399483145082626048000

Offset: 0

Peter Luschny, Jun 04 2024

nonn

Cf. A000111, A000142, A373434.

A373433 := n -> ifelse(n = 0, 1, n! * 2^n * abs(euler(n, 1/2) + euler(n, 1))):
seq(A373433(n), n = 0..18);

A373433[n_] := 2 I^(n + 1) n! PolyLog[-n, -I]; A373433[0] := 1;
Table[A373433[n], {n, 0, 18}]

# Algorithm of Ludwig Seidel (1877).
def A373433_list(n) :
    R = []; S = []; A = {-1:0, 0:1}; k = 0; f = 1; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        R.append(Am); S.append(f*Am); f *= i + 1
    return S
print(A373433_list(18))

a(n) = n! * 2^n * |Euler(n, 1/2) + Euler(n, 1)| for n >= 1.

a(n) ~ ((2*n^2)/(Pi*e^2))^n*(8*n + 4/3).

A067067 a(n) = product of nonzero digits of n! (A000142).

Original entry on oeis.org

1, 1, 2, 6, 8, 2, 14, 20, 24, 2304, 2304, 11664, 1512, 2688, 112896, 508032, 18579456, 87091200, 11854080, 368640, 6967296, 22861440, 542126592, 1872809164800, 40633270272, 559872000, 26873856000, 8561413324800, 178362777600

Offset: 0

Amarnath Murthy, Jan 03 2002

			a(10) = 2304 as 10! = 3628800 and 3*6*2*8*8 = 2304.

Harry J. Smith, Table of n, a(n) for n = 0..200

Cf. A000142, A051801.

f[n_] := Block[{d = Sort[ IntegerDigits[n! ]] }, While[ First[d] == 0, d = Drop[d, 1]]; Return[ Apply[ Times, d]]]; Table[ f[n], {n, 0, 30} ]
Table[Times@@DeleteCases[IntegerDigits[n!],0],{n,0,30}] (* Harvey P. Dale, Jan 11 2016 *)

a(n) = {vecprod(select(x->(x!=0), digits(n!)))} \\ Harry J. Smith, May 03 2010

a(n) = A051801(n!).

More terms from Jason Earls, Harvey P. Dale and Robert G. Wilson v, Jan 04 2002

cp's OEIS Frontend

A067109 Number of occurrences of the string n in n! (A000142).

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A082765 Trinomial transform of the factorial numbers (A000142).

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A089833 a(n) = A000108(n)*(A000142(n+1)-1).

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A102409 Even triangle n!. This table read by rows gives the coefficients of sum formulas of n-th factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+3, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) * n^(i-1) / (2*k-2)!.

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A102410 Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.

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A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

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A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330.

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A280062 a(n) = A049502(A000142(n)).

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A102409 Even triangle n!. This table read by rows gives the coefficients of sum formulas of n-th factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+3, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies n! = Sum_{i=1..k+3} T(i,k) n^(i-1) / (2*k-2)!.

A102410 Odd triangle n!. This table read by rows gives the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies n! = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.