A107868 -id:A107868 - cp's OEIS frontend

A129893 a(n) = s!/(s-n)! where s = (n*(n+1)/2)+1.

1, 2, 12, 210, 7920, 524160, 53721360, 7866331200, 1556675366400, 399790821830400, 129210868410624000, 51295616536721356800, 24529502681864788608000, 13903600298770901182464000

Offset: 0

Kim Dong Seok (Go Jae Song, Nam Dae Young) from KNU (gjs0419(AT)nate.com), Jun 04 2007

Bread Shop Open!. We have a loaf of bread which has a kernel of corns irregularly inside. We cut the loaf n times getting the maximal number (s, see A000124) of pieces and distribute one piece to each of n people. The remaining pieces of bread will be the prize for the winner. The sequence gives the number of cases when n pieces are distributed to n persons.

			a(2)=12 s=4,n=2 because we can write 12=4*3.
a(3)=210 s=7,n=3 because we can write 210=7*6*5.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
N. Reading, On the structure of Bruhat Order, Ph.D. dissertation, University of Minnesota, anticipated 2002.
A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964)

Reinhard Zumkeller, Table of n, a(n) for n = 0..120
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001; Annals. Combin., 7 (2003), 1-14; see Example 3.5.
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985.

Cf. A000124, A107868, A129933.

Cf. A000142.

a129893 n = a129893_list !! n
a129893_list = 1 : zipWith div (tail fs) fs where
   fs = map a000142 a000124_list
-- Reinhard Zumkeller, Oct 03 2012

Table[s=(n(n+1))/2+1;s!/(s-n)!,{n,0,20}] (* Harvey P. Dale, Nov 15 2012 *)
#[[1]]!/(#[[1]]-#[[2]])!&/@With[{nn=20},Thread[{Accumulate[ Range[0,nn]]+ 1,Range[0,nn]}]] (* Harvey P. Dale, Sep 12 2015 *)

a(n) = sPn, where s=(n*(n+1)/2)+1.

Typo fixed in a(13) by Reinhard Zumkeller, Oct 03 2012

A107869 Column 1 of triangle A107867; a(n) = binomial( n*(n+1)/2 + n+1, n).

Original entry on oeis.org

1, 3, 15, 120, 1365, 20349, 376740, 8347680, 215553195, 6358402050, 210980549208, 7778680504140, 315502265971620, 13961746143269400, 669413654240461560, 34569147570568156800, 1912924003884628655655, 112923282067713332728110

Offset: 0

Paul D. Hanna, Jun 04 2005

nonn

Cf. A107867, A107868.

PARI
```
a(n)=binomial(n*(n+1)/2+n+1,n)
```

[binomial(binomial(binomial(n+1,n),n-1),n-1) for n in range(1, 19)] # Zerinvary Lajos, Nov 30 2009

A129933 a(n) = (n*(n+1)/2+1)!/n!.

Original entry on oeis.org

1, 2, 12, 840, 1663200, 174356582400, 1561112121913344000, 1754317855900734514790400000, 341362923889542287855059017400320000000, 15163751542692044063740921044974247390216192000000000, 195932150519417838363658963697014150610807423633478962380800000000000

Offset: 0

N. J. A. Sloane, Jun 06 2007

nonn

Cf. A000124, A107868, A129893.

Table[(n (n+1)/2+1)!/n!,{n,0,20}] (* Harvey P. Dale, Aug 04 2022 *)

A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 10, 20, 1, 7, 28, 84, 210, 1, 11, 66, 286, 1001, 3003, 1, 16, 136, 816, 3876, 15504, 54264, 1, 22, 253, 2024, 12650, 65780, 296010, 1184040, 1, 29, 435, 4495, 35960, 237336, 1344904, 6724520, 30260340, 1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135

Offset: 0

Roger L. Bagula, Apr 20 2010

			Square array of T(n, k):
  1,  1,   1,    1,     1,     1,      1 ...
  1,  1,   1,    1,     1,     1,      1 ... A000012;
  1,  2,   3,    4,     5,     6,      7 ... A000027;
  1,  4,  10,   20,    35,    56,     84 ... A000292;
  1,  7,  28,   84,   210,   462,    924 ... A000579;
  1, 11,  66,  286,  1001,  3003,   8008 ... A001287;
  1, 16, 136,  816,  3876, 15504,  54264 ... A010968;
  1, 22, 253, 2024, 12650, 65780, 296010 ... A010974;
Triangle begins as:
  1;
  1,  1;
  1,  2,   3;
  1,  4,  10,   20;
  1,  7,  28,   84,   210;
  1, 11,  66,  286,  1001,   3003;
  1, 16, 136,  816,  3876,  15504,   54264;
  1, 22, 253, 2024, 12650,  65780,  296010,  1184040;
  1, 29, 435, 4495, 35960, 237336, 1344904,  6724520,  30260340;
  1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135;

G. C. Greubel, Rows n = 0..50 of the triangle, flatten

Cf. A107868 (rows sums), A158498.

[Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

T[n_, k_]= Binomial[Binomial[n, 2] + k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

row(n) = vector(n+1, k, k--; binomial(binomial(n,2) + k, k)); \\ Michel Marcus, Jul 10 2021

flatten([[binomial(binomial(n,2) +k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

T(n, k) = binomial(binomial(n, 2) + k, k).

Sum_{k=0..n} T(n, k) = A107868(n).

cp's OEIS Frontend

A129893 a(n) = s!/(s-n)! where s = (n*(n+1)/2)+1.

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A107869 Column 1 of triangle A107867; a(n) = binomial( n*(n+1)/2 + n+1, n).

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A129933 a(n) = (n*(n+1)/2+1)!/n!.

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A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

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