A157109 Triangle, read by rows, T(n, k) = binomial(n*binomial(n, floor((n-k)/2)), k).

1, 1, 1, 1, 2, 1, 1, 9, 3, 1, 1, 16, 120, 4, 1, 1, 50, 300, 2300, 5, 1, 1, 90, 4005, 7140, 58905, 6, 1, 1, 245, 10731, 518665, 211876, 1906884, 7, 1, 1, 448, 100128, 1848224, 102114376, 7624512, 74974368, 8, 1, 1, 1134, 285390, 71728020, 450710001, 28845440064, 324540216, 3477216600, 9, 1

Offset

0,5

Comments

Row sums are: {1, 2, 4, 14, 142, 2657, 70148, 2648410, 186662066, 33169921436, 11592123179902, ...}.

Links

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

Formula

T(n, k) = binomial(n*binomial(n, floor((n-k)/2)), k).

Example

Triangle begins as:
  1;
  1,   1;
  1,   2,      1;
  1,   9,      3,       1;
  1,  16,    120,       4,         1;
  1,  50,    300,    2300,         5,       1;
  1,  90,   4005,    7140,     58905,       6,        1;
  1, 245,  10731,  518665,    211876, 1906884,        7, 1;
  1, 448, 100128, 1848224, 102114376, 7624512, 74974368, 8, 1;

Maple

seq(seq( binomial(n*binomial(n, floor((n-k)/2)), k), k=0..n), n=0..10); # G. C. Greubel, Nov 30 2019

Mathematica

Table[Binomial[n*Binomial[n, Floor[(n-m)/2]], m], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = binomial(n*binomial(n, (n-k)\2), k); \\ G. C. Greubel, Nov 30 2019
(Magma) [Binomial(n*Binomial(n, Floor((n-k)/2)), k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 30 2019
(Sage) [[binomial(n*binomial(n, floor((n-k)/2)), k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 30 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> Binomial(n*Binomial(n, Int((n-k)/2)), k) ))); # G. C. Greubel, Nov 30 2019

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 23 2009

Status

approved

A185814 Exponential Riordan array (e^x,A005043(x))

1, 1, 1, 1, 2, 1, 1, 9, 3, 1, 1, 52, 30, 4, 1, 1, 545, 250, 70, 5, 1, 1, 6966, 3615, 740, 135, 6, 1, 1, 114457, 56301, 13895, 1715, 231, 7, 1, 1, 2199464, 1107148, 255416, 40390, 3416, 364, 8, 1, 1, 49219137, 24542820, 5904444, 856926, 98406, 6132, 540, 9, 1

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2013.

Formula

R(n,k) = (n!/(k-1)!)*Sum_{i=0..(n-k)} 1/i!*(Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j))/(n-i), k>0, R(n,0)=1.

Example

[1]
[1,1]
[1,2,1]
[1,9,3,1]
[1,52,30,4,1]
[1,545,250,70,5,1]
[1,6966,3615,740,135,6,1]
[1,114457,56301,13895,1715,231,7,1]

Mathematica

r[n_, 0] := 1; r[n_, k_] := (n!/(k - 1)!)*Sum[(1/i!)*Sum[Binomial[2*j - k - 1, j - 1]*(-1)^(n - j - i)*Binomial[n - i, j], {j, k, n - i}]/(n - i), {i, 0, n - k}]; Table[r[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Jul 14 2017 *)

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 05 2011

Status

approved

A174553 Triangle read by rows:t(n,m)=Sum[StirlingS2[n, k]*Eulerian[n - k + 1, m]*(-1)^(n - k - m)*k!, {k, 0, n}]

1, 1, 1, 1, 2, 1, 1, -9, 3, 1, 1, -56, -114, 4, 1, 1, -55, -590, -770, 5, 1, 1, 426, 6735, -2920, -4185, 6, 1, 1, -245, 47733, 216923, -2653, -20391, 7, 1, 1, -16372, -451052, 562016, 4011910, 109676, -93212, 8, 1, 1, -1011, -2697444, -49492896, -15614034

Offset

0,5

Comments

Row sums are:

{1, 2, 4, -4, -164, -1408, 64, 241376, 4122976, -8411008,...}.

Links

Table of n, a(n) for n=0..49.

Formula

t(n,m)=Sum[StirlingS2[n, k]*Eulerian[n - k + 1, m]*(-1)^(n - k - m)*k!, {k, 0, n}]

Example

{1},
{1, 1},
{1, 2, 1},
{1, -9, 3, 1},
{1, -56, -114, 4, 1},
{1, -55, -590, -770, 5, 1},
{1, 426, 6735, -2920, -4185, 6, 1},
{1, -245, 47733, 216923, -2653, -20391, 7, 1},
{1, -16372, -451052, 562016, 4011910, 109676, -93212, 8, 1},
{1, -1011, -2697444, -49492896, -15614034, 58337946, 1465644, -409224, 9, 1}

Mathematica

<< DiscreteMath`Combinatorica`
t[n_, m_] = Sum[StirlingS2[n, k]*Eulerian[n - k + 1, m]*(-1)^( n - k - m)*k!, {k, 0, n}];
Table[Table[t[n, m], {m, 0, n - 1}], {n, 1, 10}];
Flatten[%]

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 22 2010

Status

approved