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 A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 18
 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 4, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 4, 7, 8, 7, 5, 2, 1, 1, 1, 5, 10, 12, 11, 8, 5, 2, 1, 1, 1, 5, 11, 16, 17, 13, 9, 5, 2, 1, 1, 1, 6, 15, 23, 27, 24, 16, 10, 5, 2, 1, 1, 1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Also the number T(n,k) of (binary) min-heaps on n elements from the set {0,1} containing exactly k 1's. The sequence of column k satisfies a linear recurrence with constant coefficients of order A063915(k). LINKS Alois P. Heinz, Rows n = 0..200, flattened Eric Weisstein's World of Mathematics, Heap Wikipedia, Binary heap FORMULA Sum_{k=0..n} k * T(n,k) = A309051(n). Sum_{k=0..n} (n-k) * T(n,k) = A309052(n). Sum_{k=0..2^n-1} T(2^n-1,k) = A003095(n+1). Sum_{k=0..2^n-1} (2^n-1-k) * T(2^n-1,k) = A024358(n). Sum_{k=0..n} (T(n,k) - T(n-1,k)) = A168542(n). T(m,m-n) = A000108(n) for m >= 2^n-1 = A000225(n). T(2^n-1,k) = A202019(n+1,k+1). EXAMPLE T(6,0) = 1: 111111. T(6,1) = 3: 111011, 111101, 111110. T(6,2) = 4: 110110, 111001, 111010, 111100. T(6,3) = 4: 101001, 110010, 110100, 111000. T(6,4) = 2: 101000, 110000. T(6,5) = 1: 100000. T(6,6) = 1: 000000. T(7,4) = T(7,7-3) = A000108(3) = 5: 1010001, 1010010, 1100100, 1101000, 1110000. Triangle T(n,k) begins: 1; 1, 1; 1, 1, 1; 1, 2, 1, 1; 1, 2, 2, 1, 1; 1, 3, 3, 2, 1, 1; 1, 3, 4, 4, 2, 1, 1; 1, 4, 6, 6, 5, 2, 1, 1; 1, 4, 7, 8, 7, 5, 2, 1, 1; 1, 5, 10, 12, 11, 8, 5, 2, 1, 1; 1, 5, 11, 16, 17, 13, 9, 5, 2, 1, 1; 1, 6, 15, 23, 27, 24, 16, 10, 5, 2, 1, 1; 1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1; ... MAPLE b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand( x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)): seq(T(n), n=0..14); MATHEMATICA b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*b[n - 1 - f]]][Min[g - 1, n - g/2]]][2^Floor[Log[2, n]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]]; T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A110654, A114220 (for n>0), A326504, A326505, A326506, A326507, A326508, A326509, A326510, A326511. Row sums give A091980(n+1). T(2n,n) gives A309050. Rows reversed converge to A000108. Cf. A000225, A000295, A003095, A024358, A056971, A063915, A137560, A168542, A202019, A309051, A309052. Sequence in context: A008284 A114088 A208245 * A274190 A322596 A037306 Adjacent sequences: A309046 A309047 A309048 * A309050 A309051 A309052 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 09 2019 STATUS approved

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Last modified December 7 19:19 EST 2022. Contains 358669 sequences. (Running on oeis4.)