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A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.

%I A349706 #20 Apr 16 2025 09:02:41
%S A349706 1,0,2,0,1,4,0,1,4,8,0,1,6,12,16,0,1,10,24,32,32,0,1,18,54,80,80,64,0,
%T A349706 1,34,132,224,240,192,128,0,1,66,342,680,800,672,448,256,0,1,130,924,
%U A349706 2192,2880,2592,1792,1024,512,0,1,258,2574,7400,11000,10752,7840,4608,2304,1024
%N A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.
%H A349706 Seiichi Manyama, <a href="/A349706/b349706.txt">Antidiagonals n = 0..139, flattened</a>
%H A349706 Renate Golombek, <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1994%3A49#135">Aufgabe 1088</a>, El. Math., 49 (1994), 126-127.
%H A349706 Simsek Yilmaz, <a href="https://doi.org/10.2298/AADM1801001S">New families of special numbers for computing negative order Euler numbers and related numbers and polynomials</a>, Applicable Analysis and Discrete Mathematics 2018 Volume 12, Issue 1, Pages: 1-35. See B(n,k).
%e A349706 Square array begins:
%e A349706   1 2  4   8   16    32
%e A349706   0 1  4  12   32    80
%e A349706   0 1  6  24   80   240
%e A349706   0 1 10  54  224   800
%e A349706   0 1 18 132  680  2880
%e A349706   0 1 34 342 2192 11000
%t A349706 T[n_, k_] := Sum[Binomial[k, j] * If[j == n == 0, 1, j^n], {j, 0, k}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Nov 26 2021 *)
%o A349706 (PARI) T(n,k) = sum(j=0, k, binomial(k,j)*j^n);
%Y A349706 Rows n=0..7 give A000079, A001787, A001788, A058645, A058649, A059338, A056468, A084641.
%Y A349706 Main diagonal gives A072034.
%Y A349706 Cf. A209849.
%K A349706 nonn,tabl
%O A349706 0,3
%A A349706 _Michel Marcus_, Nov 26 2021