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 A158497 #12 Mar 19 2025 10:14:44
%S A158497 1,1,1,1,2,2,1,3,6,12,1,4,12,36,108,1,5,20,80,320,1280,1,6,30,150,750,
%T A158497 3750,18750,1,7,42,252,1512,9072,54432,326592,1,8,56,392,2744,19208,
%U A158497 134456,941192,6588344,1,9,72,576,4608,36864,294912,2359296,18874368,150994944,1,10,90,810,7290,65610,590490,5314410,47829690,430467210,3874204890
%N A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.
%C A158497 Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
%C A158497 An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
%C A158497 We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
%C A158497 For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
%C A158497 The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
%C A158497 Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.
%H A158497 G. C. Greubel, <a href="/A158497/b158497.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A158497 T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.
%F A158497 Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.
%e A158497 Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
%e A158497 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
%e A158497 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
%e A158497 1, 2, 2, 2, 2, 2, 2, 2, 2, ... A040000;
%e A158497 1, 3, 6, 12, 24, 48, 96, 192, 384, ... A003945;
%e A158497 1, 4, 12, 36, 108, 324, 972, 2916, 8748, ... A003946;
%e A158497 1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, ... A003947;
%e A158497 1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, ... A003948;
%e A158497 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, ... A003949;
%e A158497 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, ... A003950;
%e A158497 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, ... A003951;
%e A158497 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, ... A003952;
%e A158497 1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
%e A158497 1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
%e A158497 1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
%e A158497 ... ;
%e A158497 The triangle begins as:
%e A158497 1
%e A158497 1, 1;
%e A158497 1, 2, 2;
%e A158497 1, 3, 6, 12;
%e A158497 1, 4, 12, 36, 108;
%e A158497 1, 5, 20, 80, 320, 1280;
%e A158497 1, 6, 30, 150, 750, 3750, 18750;
%e A158497 1, 7, 42, 252, 1512, 9072, 54432, 326592;
%e A158497 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
%e A158497 ...;
%e A158497 T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).
%t A158497 A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
%t A158497 Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 18 2025 *)
%o A158497 (Magma)
%o A158497 A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
%o A158497 [A158497(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 18 2025
%o A158497 (SageMath)
%o A158497 def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
%o A158497 print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 18 2025
%Y A158497 Array rows n: A170733 (n=14), ..., A170769 (n=50).
%Y A158497 Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
%Y A158497 Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
%Y A158497 Sums: (-1)^n*A065440(n) (signed row).
%Y A158497 Cf. A000045, A000108, A000110, A007318, A079901, A158498.
%K A158497 nonn,tabl
%O A158497 0,5
%A A158497 _Thomas Wieder_, Mar 20 2009
%E A158497 Edited by _R. J. Mathar_, Mar 31 2009
%E A158497 More terms added by _G. C. Greubel_, Mar 18 2025