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Coefficients of a special case of Poisson-Charlier polynomials.
17

%I #35 Jul 31 2024 09:08:45

%S 1,1,1,1,3,1,1,6,8,1,1,10,29,24,1,1,15,75,145,89,1,1,21,160,545,814,

%T 415,1,1,28,301,1575,4179,5243,2372,1,1,36,518,3836,15659,34860,38618,

%U 16072,1,1,45,834,8274,47775,163191,318926,321690,125673,1,1,55,1275,16290,125853,606417,1809905,3197210,2995011,1112083,1

%N Coefficients of a special case of Poisson-Charlier polynomials.

%C Diagonals: A000012, A000217; A000012, A002104. - _Philippe Deléham_, Jun 12 2004

%C The sequence a(n) = Sum_{k = 0..n} T(n,k)*x^(n-k) is the binomial transform of the sequence b(n) = (n+x-1)! / (x-1)!. - _Philippe Deléham_, Jun 18 2004

%H G. C. Greubel, <a href="/A046716/b046716.txt">Rows n = 0..50 of the triangle, flattened</a>

%H E. A. Enneking and J. C. Ahuja, <a href="http://www.fq.math.ca/Scanned/14-1/enneking.pdf">Generalized Bell numbers</a>, Fib. Quart., 14 (1976), 67-73.

%H C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

%F Enneking and Ahuja reference gives the recurrence t(n, k) = t(n-1, k) - n*t(n-1, k-1) - (n-1)*t(n-2, k-2), with t(n, 0) = 1 and t(n, n) = (-1)^n. This sequence is T(n, k) = (-1)^k * t(n, k).

%F Sum_{k = 0..n} T(n, k)*x^(n-k) = A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively.

%F Sum_{k = 0..n} T(n, k)*2^k = A081367(n). - _Philippe Deléham_, Jun 12 2004

%F Let P(x, n) = Sum_{k = 0..n} T(n, k)*x^k, then Sum_{n>=0} P(x, n)*t^n / n! = exp(xt)/(1-xt)^(1/x). - _Philippe Deléham_, Jun 12 2004

%F T(n, 0) = 1, T(n, k) = (-1)^k * Sum_{i=n-k..n} (-1)^i*C(n, i)*S1(i, n-k), where S1 = Stirling numbers of first kind (A008275).

%F From _G. C. Greubel_, Jul 31 2024: (Start)

%F T(n, k) = T(n-1, k) + n*T(n-1, k-1) - (n-1)*T(n-2, k-2), with T(n, 0) = T(n, n) = 1.

%F Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^(n+1)*A023443(n). (End)

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 6, 8, 1;

%e 1, 10, 29, 24, 1;

%e 1, 15, 75, 145, 89, 1;

%e 1, 21, 160, 545, 814, 415, 1;

%e 1, 28, 301, 1575, 4179, 5243, 2372, 1;

%e 1, 36, 518, 3836, 15659, 34860, 38618, 16072, 1;

%p a := proc(n,k) option remember;

%p if k = 0 then 1

%p elif k < 0 then 0

%p elif k = n then (-1)^n

%p else a(n-1,k) - n*a(n-1,k-1) - (n-1)*a(n-2,k-2) fi end:

%p A046716 := (n,k) -> abs(a(n,k));

%p seq(seq(A046716(n,k),k=0..n),n=0..9); # _Peter Luschny_, Apr 05 2011

%t t[_, 0] = 1; t[n_, k_] := (-1)^k*Sum[(-1)^i*Binomial[n, i]*StirlingS1[i, n-k], {i, n-k, n}]; Table[t[n, k] // Abs, {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 10 2014 *)

%t T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1,k] +n*T[n-1,k-1] - (n-1)*T[n-2,k-2]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 31 2024 *)

%o (Magma)

%o A046716:= func< n,k | (&+[(-1)^j*Binomial(n,k-j)*StirlingFirst(j+n-k, n-k): j in [0..k]]) >;

%o [A046716(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 31 2024

%o (SageMath)

%o def A046716(n, k): return sum(binomial(n, k-j)*stirling_number1(j+n-k, n-k) for j in range(k+1))

%o flatten([[A046716(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 31 2024

%Y Diagonals include: A000012, A000217, A002104.

%Y Sums include: A000522 (row), A001339, A023443 (alternating sign row), A082030, A081367.

%K nonn,tabl,easy

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Jun 15 2004