A188137 Riordan array (1, x*(1-x)/(1-3*x+x^2)).

1, 2, 1, 5, 4, 1, 13, 14, 6, 1, 34, 46, 27, 8, 1, 89, 145, 107, 44, 10, 1, 233, 444, 393, 204, 65, 12, 1, 610, 1331, 1371, 854, 345, 90, 14, 1, 1597, 3926, 4607, 3336, 1620, 538, 119, 16, 1, 4181, 11434, 15045, 12390, 6997, 2799, 791, 152, 18, 1

Offset

1,2

Comments

The column of index 0 contains a 1 followed by zeros and is not reproduced in this triangle.

The second argument of the array definition is A(x) = A000045(x/(1-x)) = A001519(x)-1.

Triangle T(n,k), 1 <= k <= n, given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 26 2012

Links

Alois P. Heinz, Rows n = 1..141, flattened

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(n,m) = Sum_{k=m..n} binomial(n-1,k-1) * Sum_{i=ceiling((k-m)/2)..k-m} binomial(i,k-m-i)*binomial(m+i-1,m-1), 0<m<=n.

T(n,m) = Sum_{i=1..n-m+1} A001519(i)*T(n-i,m-1).

T(n,1) = A001519(n).

Sum_{m=1..n} T(n,m) = A007052(n-1).

G.f.: (1-3x+x^2)/(1-(3+y)*x + (1+y)*x^2). - Philippe Deléham, Jan 26 2012

Example

Triangle begins:
   1;
   2,   1;
   5,   4,   1;
  13,  14,   6,  1;
  34,  46,  27,  8,  1;
  89, 145, 107, 44, 10, 1;
From Philippe Deléham, Jan 26 2012: (Start)
Triangle (0,2,1/2,1/2,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,...) begins:
  1;
  0,  1;
  0,  2,   1;
  0,  5,   4,   1;
  0, 13,  14,   6,  1;
  0, 34,  46,  27,  8,  1;
  0, 89, 145, 107, 44, 10, 1; (End)

Maple

A188137 := proc(n, m) add( binomial(n-1, k-1) *add(binomial(i, k-m-i) *binomial(m+i-1, m-1), i=ceil((k-m)/2)..k-m), k=m..n) ; end proc:
seq(seq(A188137(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Mar 30 2011

Mathematica

t[n_, m_] := Sum[ Binomial[n - 1, k - 1]*Sum[ Binomial[i, k - m - i]*Binomial[m + i - 1, m - 1], {i, Ceiling[(k - m)/2], k - m}], {k, m, n}]; Table[t[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)

Prog

(Maxima)
T(n, m):=sum(binomial(n-1, k-1) *sum(binomial(i, k-m-i) *binomial(m+i-1, m-1), i, ceiling((k-m)/2), k-m), k, m, n);

Crossrefs

Cf. A001519 (column 1), A030267 (column 2).

Sequence in context: A096164 A201166 A318942 * A201165 A171488 A171651

Adjacent sequences: A188134 A188135 A188136 * A188138 A188139 A188140

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 21 2011

Status

approved