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

0,11

Comments

Same as A187080, with trailing zeros omitted.

Links

Alois P. Heinz, Rows n = 0..1000, flattened

Example

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;

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 *)

Prog

(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

Crossrefs

Row sums give A005169.

Columns 0-2 give A000007, A000012, A000071.

Cf. A047998, A187080.

Sequence in context: A218581 A307177 A340264 * A131487 A230747 A308628

Adjacent sequences: A291875 A291876 A291877 * A291879 A291880 A291881

Keyword

nonn,tabf

Author

Seiichi Manyama, Sep 05 2017

Status

approved