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 A193913 Diagonal element T(n,n) of the infinite array with T(n,1) = T(1,n) = Fibonacci(n) and recursively T(n,k) = T(n-1,k-1) + T(n,k-1) + T(n-1,k). 1
 1, 3, 15, 79, 425, 2317, 12749, 70631, 393379, 2200203, 12348645, 69507969, 392211153, 2217824883, 12564291759, 71294454543, 405135974649, 2305189276605, 13131574749357, 74883034577575, 427430124521651, 2441889639394043, 13961588736578245, 79884779408549249 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1309 FORMULA T(n,1) = T(1,n) = A000045(n). T(n,k) = 0 if n <= 0 or k <= 0. T(n,k) = T(n-1,k-1) + T(n,k-1) + T(n-1,k), n > 1, k > 1. T(n,k) = T(k,n). T(n,2) = A001911(n). T(n,3) = A000045(n+6) - A004767(n+1). - R. J. Mathar, Aug 28 2011 T(n,4) = A000045(n+9) - 34 - 14*n - 4*n^2. T(n,5) = A000045(n+12) -8*n^3/3 -14*n^2 -208*n/3 -142. - R. J. Mathar, Aug 29 2011 EXAMPLE Diagonal of the matrix T(n,k) which starts for n,k >= 1 as: 1 1 2 3 5 8 13 21 34 55 1 3 6 11 19 32 53 87 142 231 2 6 15 32 62 113 198 338 567 940 3 11 32 79 173 348 659 1195 2100 3607 5 19 62 173 425 946 1953 3807 7102 12809 8 32 113 348 946 2317 5216 10976 21885 41796 13 53 198 659 1953 5216 12749 28941 61802 125483 21 87 338 1195 3807 10976 28941 70631 161374 348659 MAPLE A := proc(n, k) option remember; if n<=0 or k<=0 then 0; elif k = 1 then combinat[fibonacci](n) ; elif n = 1 then combinat[fibonacci](k) ; else procname(n-1, k-1)+procname(n, k-1)+procname(n-1, k) ; end if; end proc: A193913 := proc(n) A(n, n) ; end proc: # R. J. Mathar, Aug 28 2011 # second Maple program: b:= proc(x, y) option remember; `if`(x<2, (<<0|1>, <1|1>>^y)[1, 2], b(x-1, y)+b(sort([x, y-1])[])+b(x-1, y-1)) end: a:= n-> b(n\$2): seq(a(n), n=1..29); # Alois P. Heinz, Jul 14 2021 MATHEMATICA T[n_ /; n>=1, 1] := T[1, n] = Fibonacci[n]; T[n_ /; n>=1, k_] /; n>=k := T[n, k] = T[n-1, k-1] + T[n, k-1] + T[n-1, k]; T[n_, k_] /; k>n := T[k, n]; T[_, _] = 0; a[n_] := T[n, n]; Array[a, 24] (* Jean-François Alcover, Apr 26 2022 *) PROG (MATLAB) function [ out ] = a( n ) ary=zeros(n, n); ary(1, 1)=1; if(n==1) out= 1; return; end ary(2, 1)=1; ary(1, 2)=1; for i=3:n ary(i, 1)=ary(i-1, 1)+ary(i-2, 1); ary(1, i)=ary(1, i-1)+ary(1, i-2); end for i=2:n for j=2:n ary(i, j)=ary(i, j-1)+ary(i-1, j-1)+ary(i-1, j); end end out=ary(n, n) CROSSREFS Cf. A000045. Cf. A001911, A004767. Sequence in context: A031884 A337284 A186942 * A292097 A052755 A137957 Adjacent sequences: A193910 A193911 A193912 * A193914 A193915 A193916 KEYWORD nonn AUTHOR Jaley H Dholakiya, Aug 09 2011 STATUS approved

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Last modified March 5 02:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)