A168422 Number triangle with row sums given by quadruple factorial numbers A001813.

1, 1, 1, 7, 4, 1, 71, 39, 9, 1, 1001, 536, 126, 16, 1, 18089, 9545, 2270, 310, 25, 1, 398959, 208524, 49995, 7120, 645, 36, 1, 10391023, 5394991, 1301139, 190435, 18445, 1197, 49, 1, 312129649, 161260336, 39066076, 5828704, 589750, 41776, 2044, 64, 1

Offset: 0

Author

Paul Barry, Nov 25 2009

Keywords

• easy
• nonn
• tabl

Comments

Reversal of coefficient array for the polynomials P(n,x) = Sum_{k=0..n} (C(n+k,2k)*(2k)!/k!)*x^k*(1-x)^(n-k).

Note that P(n,x) = Sum_{k=0..n} A113025(n,k)*x^k*(1-x)^(n-k). Row sums are A001813.

Examples

			Triangle begins
          1
          1         1
          7         4        1
         71        39        9       1
       1001       536      126      16      1
      18089      9545     2270     310     25     1
     398959    208524    49995    7120    645    36    1
   10391023   5394991  1301139  190435  18445  1197   49  1
  312129649 161260336 39066076 5828704 589750 41776 2044 64 1
Production matrix begins
        1       1
        6       3       1
       40      20       5      1
      336     168      42      7     1
     3456    1728     432     72     9    1
    42240   21120    5280    880   110   11   1
   599040  299520   74880  12480  1560  156  13  1
  9676800 4838400 1209600 201600 25200 2520 210 15 1
Complete this with a top row (1,0,0,0,...) and invert: we get
    1
   -1   1
   -3  -3   1
   -5  -5  -5   1
   -7  -7  -7  -7   1
   -9  -9  -9  -9  -9   1
  -11 -11 -11 -11 -11 -11   1
  -13 -13 -13 -13 -13 -13 -13   1
  -15 -15 -15 -15 -15 -15 -15 -15   1
  -17 -17 -17 -17 -17 -17 -17 -17 -17   1
		

Links

• William P. Orrick, Table of n, a(n) for n = 0..10010

Crossrefs

Column 1 is |A002119|.

Sum_{k=0..n} T(n,k) * 2^k, is A001517(n).

Cf. A079267.

Programs

PARI

T(n,k)={sum(j=k, n, (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!))/k!} \\ Andrew Howroyd, Mar 24 2023

SageMath

def T(n,k):
    return(sum((-1)^(j-k) * binomial(2*n-j,n) * binomial(n,j)\
     * binomial(j,k) * factorial(n-j)\
     for j in range(k,n+1))) # William P. Orrick, Mar 24 2023

Formula

T(n,k) = (1/k!)*Sum_{j=k..n} (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!).

Extensions

Corrected and extended by William P. Orrick, Mar 24 2023