OFFSET
0,3
REFERENCES
S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
LINKS
Vincenzo Librandi, Rows n = 0..50, flattened
S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
FORMULA
a(p, n) = Sum_{k=-1..n} Lp(n, k) where Lp(n, k) satisfies the recurrence: Lp(n, k) = Sum_{i=k..n} L(p-1, n, i)*L(p, i-1, k-1) for p, n >= 1, k >= 0 with the boundary conditions: Lp(n, n) = Lp(n, -1) = 1 for all p >= 1, n >= -1; L0(n, n) = L0(n, -1) = 1 and L0(n, k) = 0 for k different from -1 or n. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 10 2005
EXAMPLE
Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 3, 2;
1, 2, 4, 4, 2;
1, 2, 5, 8, 5, 2;
1, 2, 6, 15, 16, 6, 2;
1, 2, 7, 26, 52, 32, 7, 2;
1, 2, 8, 42, 152, 203, 64, 8, 2;
1, 2, 9, 64, 392, 1144, 877, 128, 9, 2;
1, 2, 10, 93, 904, 5345, 10742, 4140, 256, 10, 2;
... - Vincenzo Librandi, Jul 24 2013
MAPLE
L:=proc(p, n, k) options remember: local i: if (k=-1 or k=n) and n>=-1 and p>=1 then RETURN(1) elif p=0 and (k=-1 or k=n) then RETURN(1) elif p=0 and (k<>-1 and k<>n) then RETURN(0) elif p>=1 and n>=1 then RETURN(add(L(p-1, n, i)*L(p, i-1, k-1), i=k..n)) fi: end; M:=(p, n)->add(L(p, n, k), k=-1..n); seq(seq(M(n-i+1, i-1), i=0..n+1), n=-1..12); # first method (Pab Ter)
L:=proc(p, n, k) options remember: local i: if (k=-1 or k=n) and n>=-1 and p>=1 then RETURN(1) elif p=0 and (k=-1 or k=n) then RETURN(1) elif p=0 and (k<>-1 and k<>n) then RETURN(0) elif p>=1 and n>=1 then RETURN(add(L(p-1, n, i)*L(p, i-1, k-1), i=k..n)) fi: end; M:=proc(p, n) options remember: local i: if n<1 and n>-2 and p>=0 then RETURN([1, 2][n+2]) elif p=0 and n>=0 then RETURN(2) elif p>=1 and n>=1 then RETURN(1+add(L(p-1, n, i)*M(p, i-1), i=0..n)) fi: end; seq(seq(M(n-i+1, i-1), i=0..n+1), n=-1..12); # 2nd method (Pab Ter)
MATHEMATICA
a[p_, n_] := Sum[ lp[p, n, k], {k, -1, n}]; lp[p_ /; p >= 1, n_ /; n >= 1, k_ /; k >= 0] := lp[p, n, k] = Sum[ lp[p-1, n, i] lp[p, i-1, k-1], {i, k, n}]; lp[p_ /; p >= 1, n_ /; n >= -1, n_ /; n >= -1] := lp[p, n, n] = 1; lp[p_ /; p >= 1, n_ /; n >= -1, -1] := lp[p, n, -1] = 1; lp[0, n_, n_] := lp[0, n, n] = 1; lp[0, n_, -1] := lp[0, n, -1] = 1; lp[0, n_, k_] /; k != -1 && k != n := lp[0, n, k] = 0; m[p_, n_] := m[p, n] = Sum[ lp[p, n, k], {k, -1, n}]; row[n_] := Table[ m[n-i, i-1], {i, 0, n}]; Flatten[ Table[ row[n], {n, 0, 11}]] (* Jean-François Alcover, Dec 16 2011, after Pab Ter *)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 10 2005
STATUS
approved