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

0,4

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.

Links

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

Formula

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

Example

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

Prog

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

Crossrefs

Column 1 is |A002119|.

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

Cf. A079267.

Sequence in context: A258330 A011407 A021907 * A187056 A103240 A155531

Adjacent sequences: A168419 A168420 A168421 * A168423 A168424 A168425

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 25 2009

Extensions

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

Status

approved