A291878 Triangle read by rows: T(n,k) = number of fountains of n coins and height k.
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 4, 0, 1, 7, 1, 0, 1, 12, 2, 0, 1, 20, 5, 0, 1, 33, 11, 0, 1, 54, 22, 1, 0, 1, 88, 44, 2, 0, 1, 143, 85, 5, 0, 1, 232, 161, 12, 0, 1, 376, 302, 25, 0, 1, 609, 559, 52, 1, 0, 1, 986, 1026, 105, 2, 0, 1, 1596, 1870, 207, 5, 0, 1
Offset: 0
Examples
T(6, 1) = 1; . O O O O O O . ------------------------------------------------------- T(6, 2) = 7; .. O O ......... O O ..... O . O .. . O O O O ... O O O O ... O O O O . ....................................................... .. O ............. O ............. O ............. O .. . O O O O O ... O O O O O ... O O O O O ... O O O O O . ------------------------------------------------------- T(6, 3) = 1; ... O ... .. O O .. . O O O . ------------------------------------------------------- First few rows are: 1; 0, 1; 0, 1; 0, 1, 1; 0, 1, 2; 0, 1, 4; 0, 1, 7, 1; 0, 1, 12, 2; 0, 1, 20, 5; 0, 1, 33, 11; 0, 1, 54, 22, 1; 0, 1, 88, 44, 2;
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
Programs
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Maple
b:= proc(n, i, h) option remember; `if`(n=0, x^h, add(b(n-j, j, max(h, j)), j=1..min(i+1, n))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)): seq(T(n), n=0..30); # Alois P. Heinz, Sep 05 2017 -
Mathematica
b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]]; T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]& @ b[n, 0, 0]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, May 31 2019, after Alois P. Heinz *) -
Python
from sympy.core.cache import cacheit from sympy import Symbol, Poly, flatten x=Symbol('x') @cacheit def b(n, i, h): return x**h if n==0 else sum([b(n - j, j, max(h, j)) for j in range(1, min(i + 1, n) + 1)]) def T(n): return 1 if n==0 else Poly(b(n, 0, 0)).all_coeffs()[::-1] print(flatten(map(T, range(31)))) # Indranil Ghosh, Sep 06 2017
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