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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A155863 A173882 A174045

Adjacent sequences:  A156136 A156137 A156138 * A156140 A156141 A156142

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 04 2009

STATUS

approved

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Last modified March 26 22:42 EDT 2019. Contains 321565 sequences. (Running on oeis4.)