A168422 - OEIS

%I #16 Mar 31 2023 14:41:04

%S 1,1,1,7,4,1,71,39,9,1,1001,536,126,16,1,18089,9545,2270,310,25,1,

%T 398959,208524,49995,7120,645,36,1,10391023,5394991,1301139,190435,

%U 18445,1197,49,1,312129649,161260336,39066076,5828704,589750,41776,2044,64,1

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

%C 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).

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

%H William P. Orrick, <a href="/A168422/b168422.txt">Table of n, a(n) for n = 0..10010</a>

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

%e Triangle begins

%e           1

%e           1         1

%e           7         4        1

%e          71        39        9       1

%e        1001       536      126      16      1

%e       18089      9545     2270     310     25     1

%e      398959    208524    49995    7120    645    36    1

%e    10391023   5394991  1301139  190435  18445  1197   49  1

%e   312129649 161260336 39066076 5828704 589750 41776 2044 64 1

%e Production matrix begins

%e         1       1

%e         6       3       1

%e        40      20       5      1

%e       336     168      42      7     1

%e      3456    1728     432     72     9    1

%e     42240   21120    5280    880   110   11   1

%e    599040  299520   74880  12480  1560  156  13  1

%e   9676800 4838400 1209600 201600 25200 2520 210 15 1

%e Complete this with a top row (1,0,0,0,...) and invert: we get

%e     1

%e    -1   1

%e    -3  -3   1

%e    -5  -5  -5   1

%e    -7  -7  -7  -7   1

%e    -9  -9  -9  -9  -9   1

%e   -11 -11 -11 -11 -11 -11   1

%e   -13 -13 -13 -13 -13 -13 -13   1

%e   -15 -15 -15 -15 -15 -15 -15 -15   1

%e   -17 -17 -17 -17 -17 -17 -17 -17 -17   1

%o (SageMath)

%o def T(n,k):

%o     return(sum((-1)^(j-k) * binomial(2*n-j,n) * binomial(n,j)\

%o      * binomial(j,k) * factorial(n-j)\

%o      for j in range(k,n+1))) # _William P. Orrick_, Mar 24 2023

%o (PARI) T(n,k)={sum(j=k, n, (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!))/k!} \\ _Andrew Howroyd_, Mar 24 2023

%Y Column 1 is |A002119|.

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

%Y Cf. A079267.

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Nov 25 2009

%E Corrected and extended by _William P. Orrick_, Mar 24 2023