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 A097712 Lower triangular matrix T, read by rows, such that T(n,0) = 1 and T(n,k) = T(n-1,k) + T^2(n-1,k-1) for k>0, where T^2 is the matrix square of T. 9
 1, 1, 1, 1, 3, 1, 1, 8, 7, 1, 1, 25, 44, 15, 1, 1, 111, 346, 208, 31, 1, 1, 809, 4045, 3720, 912, 63, 1, 1, 10360, 77351, 99776, 35136, 3840, 127, 1, 1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1, 1, 9708797, 145895764, 319822055, 189724354, 37329584, 2608864, 64256, 511, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums form A016121. Column 1 forms A097713. This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093662. LINKS EXAMPLE T(5,1) = T(4,1) + T^2(4,0) = 25 + 86 = 111. T(5,2) = T(4,2) + T^2(4,1) = 44 + 302 = 346. T(5,3) = T(4,3) + T^2(4,2) = 15 + 193 = 208. Rows of T begin: [1], [1,1], [1,3,1], [1,8,7,1], [1,25,44,15,1], [1,111,346,208,31,1], [1,809,4045,3720,912,63,1], [1,10360,77351,99776,35136,3840,127,1], [1,236952,2535715,4341249,2032888,308976,15808,255,1], [1,9708797,145895764,319822055,189724354,37329584,2608864,64256,511,1],... Rows of T^2 begin: [1], [2,1], [5,6,1], [17,37,14,1], [86,302,193,30,1], [698,3699,3512,881,62,1], [9551,73306,96056,34224,3777,126,1], [226592,2458364,4241473,1997752,305136,15681,254,1], [9471845,143360049,315480806,187691466,37020608,2593056,64001,510,1],... Column 0 of T^2 forms A016121. Row sums of T^2 form the first differences of A016121. MATHEMATICA T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2019 *) PROG (PARI) T(n, k)=if(n<0 || k>n, 0, if(n==k, 1, if(k==0, 1, T(n-1, k)+sum(j=0, n-1, T(n-1, j)*T(j, k-1)); ))) CROSSREFS Cf. A016121, A093662, A097713, A097710. Sequence in context: A263859 A124469 A094816 * A238688 A174117 A157210 Adjacent sequences:  A097709 A097710 A097711 * A097713 A097714 A097715 KEYWORD nonn,tabl,changed AUTHOR Paul D. Hanna, Aug 24 2004 STATUS approved

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Last modified October 16 06:34 EDT 2019. Contains 328051 sequences. (Running on oeis4.)