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 A244523 Irregular triangle read by rows: T(n,k) is the number of identity trees with n nodes and maximal branching factor k. 10
 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 5, 0, 1, 10, 1, 0, 1, 21, 3, 0, 1, 42, 9, 0, 1, 87, 25, 0, 1, 178, 66, 2, 0, 1, 371, 170, 6, 0, 1, 773, 431, 21, 0, 1, 1630, 1076, 63, 0, 1, 3447, 2665, 185, 1, 0, 1, 7346, 6560, 512, 7, 0, 1, 15712, 16067, 1403, 26, 0, 1, 33790, 39219, 3750, 91, 0, 1, 72922, 95476, 9928, 291 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS Row sums give A004111. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 1..285, flattened EXAMPLE Triangle starts: 01: 1, 02: 0, 1, 03: 0, 1, 04: 0, 1, 1, 05: 0, 1, 2, 06: 0, 1, 5, 07: 0, 1, 10, 1, 08: 0, 1, 21, 3, 09: 0, 1, 42, 9, 10: 0, 1, 87, 25, 11: 0, 1, 178, 66, 2, 12: 0, 1, 371, 170, 6, 13: 0, 1, 773, 431, 21, 14: 0, 1, 1630, 1076, 63, 15: 0, 1, 3447, 2665, 185, 1, 16: 0, 1, 7346, 6560, 512, 7, 17: 0, 1, 15712, 16067, 1403, 26, 18: 0, 1, 33790, 39219, 3750, 91, 19: 0, 1, 72922, 95476, 9928, 291, 20: 0, 1, 158020, 231970, 25969, 885, 3, 21: 0, 1, 343494, 562736, 67462, 2588, 15, 22: 0, 1, 749101, 1363640, 174039, 7373, 70, 23: 0, 1, 1638102, 3301586, 446884, 20555, 256, 24: 0, 1, 3591723, 7988916, 1142457, 56413, 884, 25: 0, 1, 7893801, 19322585, 2911078, 152812, 2840, 3, ... The A004111(7) = 12 level-sequences and the branching sequences for the identity trees with 7 nodes are (dots for zeros), together with the maximal branching factors, are: 01: [ . 1 2 3 4 5 6 ] [ 1 1 1 1 1 1 . ] 1 02: [ . 1 2 3 4 5 4 ] [ 1 1 1 2 1 . . ] 2 03: [ . 1 2 3 4 5 3 ] [ 1 1 2 1 1 . . ] 2 04: [ . 1 2 3 4 5 2 ] [ 1 2 1 1 1 . . ] 2 05: [ . 1 2 3 4 5 1 ] [ 2 1 1 1 1 . . ] 2 06: [ . 1 2 3 4 3 2 ] [ 1 2 2 1 . . . ] 2 07: [ . 1 2 3 4 3 1 ] [ 2 1 2 1 . . . ] 2 08: [ . 1 2 3 4 2 3 ] [ 1 2 1 1 . 1 . ] 2 09: [ . 1 2 3 4 2 1 ] [ 2 2 1 1 . . . ] 2 10: [ . 1 2 3 4 1 2 ] [ 2 1 1 1 . 1 . ] 2 11: [ . 1 2 3 2 1 2 ] [ 2 2 1 . . 1 . ] 2 12: [ . 1 2 3 1 2 1 ] [ 3 1 1 . 1 . . ] 3 This gives row n=7: [0, 1, 10, 1, 0, 0, ... ]. MAPLE b:= proc(n, i, t, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(b(i-1\$2, k\$2), j)* b(n-i*j, i-1, t-j, k), j=0..min(t, n/i)))) end: g:= proc(n) local k; if n=1 then 0 else for k while T(n, k)>0 do od; k-1 fi end: T:= (n, k)-> b(n-1\$2, k\$2) -`if`(k=0, 0, b(n-1\$2, k-1\$2)): seq(seq(T(n, k), k=0..g(n)), n=1..25); MATHEMATICA b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k], j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]]; g[n_] := If[ n == 1 , 0, For[k=1, T[n, k]>0 , k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k] - If[k == 0, 0, b[n-1, n-1, k-1, k-1]]; Table[Table[T[n, k], {k, 0, g[n]}], {n, 1, 25}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *) CROSSREFS Columns k=0-10 give: A000007, A000012 (for n>0), A245747, A245748, A245749, A245750, A245751, A245752, A245753, A245754, A245755. Cf. A004111 (identity trees), A244372 (unlabeled rooted trees by outdegree). Sequence in context: A337085 A209687 A108263 * A325304 A134433 A125183 Adjacent sequences: A244520 A244521 A244522 * A244524 A244525 A244526 KEYWORD nonn,tabf AUTHOR Joerg Arndt and Alois P. Heinz, Jul 30 2014 STATUS approved

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Last modified September 26 07:08 EDT 2023. Contains 365653 sequences. (Running on oeis4.)