login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 * A309280 A155863 A173882

Adjacent sequences:  A156136 A156137 A156138 * A156140 A156141 A156142

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 04 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 26 12:37 EST 2020. Contains 338639 sequences. (Running on oeis4.)