A001720 -id:A001720 - cp's OEIS frontend

A062263 Sixth (unsigned) column of triangle A062140 (generalized a=4 Laguerre).

1, 60, 2310, 73920, 2162160, 60540480, 1664863200, 45664819200, 1261490630400, 35321737651200, 1006669523059200, 29284931579904000, 871226714502144000, 26538906072526848000, 828392996692445184000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials

Cf. A001720, A062140, A062199, A062260, A062261, A062262.

[Factorial(n+5)*Binomial(n+9, 9)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+9,9]/5!, {n, 0, 20}] (* G. c. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062263.txt", n, " ", f*binomial(n + 9, 9)/120) ) } \\ Harry J. Smith, Aug 03 2009

E.g.f.: N(4;5, x)/(1-x)^15, with N(4;5, x) := Sum_{k=0..5} A062264(5, k)* x^k = 1 + 45*x + 360*x^2 + 840*x^3 + 630*x^4 + 226*x^5.
a(n) = A062140(n+5, 5).
a(n) = (n+5)!*binomial(n+9, 9)/5!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-9) = (-1)^(n-1)*f(n,9,-6), (n>=9). - Milan Janjic, Mar 01 2009

A144893 Second column (m=2) of triangle A144891 (S1hat(5)).

Original entry on oeis.org

1, 5, 55, 360, 3630, 29820, 321300, 3225600, 38808000, 466300800, 6360379200, 89703936000, 1389213504000, 22565765376000, 394272204480000, 7248941973504000, 141496402037760000, 2901258659819520000, 62617333274496000000, 1414755114367795200000, 33446554797269053440000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144891, A001720 (first column), A144894 (third column).

a(n) = A144891(n+2,2), n>=0.

A144894 Third column (m=3) of triangle A144891 (S1hat(5)).

Original entry on oeis.org

1, 5, 55, 485, 4380, 39570, 421800, 4265100, 49455000, 594001800, 7784683200, 107814672000, 1624964544000, 25881953328000, 443107288512000, 8028628336512000, 154539996629760000, 3135393617489280000, 67045955961922560000, 1503619681171829760000, 35321502985569884160000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144891, A001720 (first column), A144893 (second column).

a(n) = A144891(n+3,3), n>=0.

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 13.

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.

Triangle A094587 with 1st column A000262.

T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
T(n,3) = A001710(n-1) for n > 2.
T(n,4) = A001715(n-1) for n > 3.
T(n,5) = A001720(n-1) for n > 4.
T(n,6) = A001725(n-1) for n > 5.
T(n,7) = A001730(n-1) for n > 6.
T(n,8) = A049388(n-8) for n > 7.
T(n,9) = A049389(n-9) for n > 8.
T(n,10) = A049398(n-10) for n > 9.
T(n,11) = A051431(n-11) for n > 10.

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A062263 Sixth (unsigned) column of triangle A062140 (generalized a=4 Laguerre).

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A144893 Second column (m=2) of triangle A144891 (S1hat(5)).

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A144894 Third column (m=3) of triangle A144891 (S1hat(5)).

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A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

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A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

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