This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081719 #34 May 27 2025 14:54:12
%S A081719 1,0,1,0,1,1,0,1,3,1,0,1,5,5,1,0,1,9,14,7,1,0,1,13,32,27,9,1,0,1,20,
%T A081719 66,80,44,11,1,0,1,28,123,203,160,65,13,1,0,1,40,222,465,486,280,90,
%U A081719 15,1,0,1,54,377,985,1305,990,448,119,17,1,0,1,75,630,1978,3203,3051,1807,672,152,19,1
%N A081719 Triangle T(n,k) read by rows, related to Faà di Bruno's formula (n >= 0 and 0 <= k <= n).
%C A081719 From _Petros Hadjicostas_, May 30 2020: (Start)
%C A081719 We may prove _Philippe Deléham_'s formula by induction on n. Let P(n,k) = A008284(n,k) and b(n) = A039809(n). For n = 0, Sum_{k=0..0} T(0,k) = 1 = b(1). Let n >= 1, and assume his formula is true for all s < n, i.e., Sum_{k=0..s} T(s,k) = b(s+1).
%C A081719 Then Sum_{k=0..n} T(n, k) = Sum_{k=1..n} T(n,k) = Sum_{k=1..n} Sum_{s=k-1..n-1} P(n+1, s+1)*T(s, k-1) = Sum_{s=0..n-1} P(n+1, s+1) Sum_{k=1..s+1} T(s, k-1) = Sum_{s=0..n-1} P(n+1, s+1) Sum_{m=0..s} T(s,m) = Sum_{s=0..n-1} P(n+1, s+1)*b(s+1) = Sum_{r=1..n} P(n+1, r)*b(r) = b(n+1) (by the definition of b = A039809). (End)
%H A081719 Warren P. Johnson, <a href="https://web.archive.org/web/20220813233953/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Johnson217-234.pdf">The curious history of Faà di Bruno's formula</a>, American Mathematical Monthly, 109 (2002), 217-234.
%H A081719 Warren P. Johnson, <a href="https://www.jstor.org/stable/2695352?seq=1">The curious history of Faà di Bruno's formula</a>, American Mathematical Monthly, 109 (2002), 217-234.
%H A081719 Winston C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Mathematics, 222(1-3), July 2000, 235-245.
%F A081719 There is a recurrence involving the partition function A008284.
%F A081719 Sum_{k=0..n} T(n,k) = A039809(n+1). - _Philippe Deléham_, Sep 30 2006
%F A081719 From _Petros Hadjicostas_, May 30 2020: (Start)
%F A081719 T(n, k) = Sum_{s=k-1..n-1} A008284(n+1, s+1)*T(s, k-1) for 1 <= k <= n with T(0,0) = 1 and T(n,0) = 0 for n >= 1.
%F A081719 T(n, k=2) = A007042(n) = A047812(n,2). (End)
%e A081719 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A081719 1;
%e A081719 0, 1;
%e A081719 0, 1, 1;
%e A081719 0, 1, 3, 1;
%e A081719 0, 1, 5, 5, 1;
%e A081719 0, 1, 9, 14, 7, 1;
%e A081719 0, 1, 13, 32, 27, 9, 1;
%e A081719 0, 1, 20, 66, 80, 44, 11, 1;
%e A081719 ...
%t A081719 (* b = A008284 *)
%t A081719 b[n_, k_]:= b[n, k]= If[n>0 && k>0, b[n-1, k-1] + b[n-k, k], Boole[n==0 && k==0]];
%t A081719 T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k==0, 0, Sum[T[j, k-1]*b[n+1, j+1], {j, k-1, n-1}] ]];
%t A081719 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 31 2020 *)
%o A081719 (PARI) P(n, k)=#partitions(n-k, k); /* A008284 */
%o A081719 tabl(nn) = {A = matrix(nn, nn, n, k, 0); A[1,1] = 1; for(n=2, nn, for(k=2, n, A[n,k] = sum(s=k-2, n-2, P(n, s+1)*A[s+1,k-1])));
%o A081719 for (n=1, nn, for (k=1, n, print1(A[n, k], ", "); ); print(); ); } \\ _Petros Hadjicostas_, May 29 2020
%Y A081719 Cf. A007042, A008284, A039809, A047812.
%K A081719 nonn,tabl,easy
%O A081719 0,9
%A A081719 _N. J. A. Sloane_, Apr 05 2003
%E A081719 More terms from _Emeric Deutsch_, Feb 28 2005