A156139 Triangle T(n,k) = (2*n-k-1)*T(n-1,k-1) + (k+1)*T(n-1,k), with T(n,1) = T(n,n) = 1, 1 <= k <= n, read by rows.

1, 1, 1, 1, 6, 1, 1, 23, 28, 1, 1, 76, 250, 145, 1, 1, 237, 1608, 2475, 876, 1, 1, 722, 8802, 26847, 25056, 6139, 1, 1, 2179, 43872, 231057, 418806, 268477, 49120, 1, 1, 6552, 205994, 1725621, 5285520, 6486205, 3077730, 442089, 1, 1, 19673, 928808, 11718015, 55871814, 114115195, 102456300, 37833831, 4420900, 1

Offset

1,5

Comments

Row sums are s(n) = 1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, ...

Links

G. C. Greubel, Rows n=1..25 of triangle, flattened

Leonard M. Smiley, Completion of a Rational Function Sequence of Carlitz, page 2.

Formula

Row sums s(n) = Sum_{k=1..n} T(n,k) seem to obey (n-2)*s(n) - (1-4*n+2*n^2)*s(n-1) + (3-5*n+2*n^2) * s(n-2)=0, n > 0. - R. J. Mathar, Jun 24 2011

Example

Triangle begins with:
1;
1,     1;
1,     6,      1;
1,    23,     28,       1;
1,    76,    250,     145,       1;
1,   237,   1608,    2475,     876,       1;
1,   722,   8802,   26847,   25056,    6139,       1;
1,  2179,  43872,  231057,  418806,  268477,   49120,      1;
1,  6552, 205994, 1725621, 5285520, 6486205, 3077730, 442089,   1;

Maple

A156139 := proc(n, k) option remember; if k= 1 or k=n then 1; else (2*n-k-1)*procname(n-1, k-1)+(k+1)*procname(n-1, k) ; end if; end proc:
seq(seq(A156139(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Jun 24 2011

Mathematica

T[n_, 1]:= 1; T[n_, n_] := 1;
T[n_, k_]:= (2*n-k-1)*T[n-1, k-1] + (k+1)*T[n-1, k];
TableForm[Table[T[n, k], {n, 10}, {k, n}], TableAlignments -> Right];
Table[Table[T[n, k], {k, n}], {n, 10}]//Flatten

Prog

(PARI) {T(n, k) = if(k==1, 1, if(k==n, 1, (2*n-k-1)*T(n-1, k-1) + (k+1)*T(n-1, k)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 25 2019

Crossrefs

Sequence in context: A138076 A060187 A174527 * A309280 A155863 A173882

Adjacent sequences: A156136 A156137 A156138 * A156140 A156141 A156142

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 04 2009

Status

approved