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A157011 Triangle T(n,k) read by rows: T(n,k)= (k-1)*T(n-1,k) + (n-k+2)*T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1. 5
1, 1, 2, 1, 5, 4, 1, 9, 23, 8, 1, 14, 82, 93, 16, 1, 20, 234, 607, 343, 32, 1, 27, 588, 2991, 3800, 1189, 64, 1, 35, 1365, 12501, 30155, 21145, 3951, 128, 1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256, 1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums are apparently in A002627.

The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

EXAMPLE

The triangle starts in row n=1 as:

1;

1, 2;

1, 5, 4;

1, 9, 23, 8;

1, 14, 82, 93, 16;

1, 20, 234, 607, 343, 32;

1, 27, 588, 2991, 3800, 1189, 64;

1, 35, 1365, 12501, 30155, 21145, 3951, 128;

1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256;

1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512;

MAPLE

A157011 := proc(n, k) if k <0 or k >= n then 0; elif k =0 then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc: # R. J. Mathar, Jun 18 2011

MATHEMATICA

e[n_, 0, m_]:= 1;

e[n_, k_, m_]:= 0 /; k >= n;

e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];

Table[Flatten[Table[Table[e[n, k, m], {k, 0, n-1}], {n, 1, 10}]], {m, 0, 10}]

T[n_, 1]:= 1; T[n_, n_]:= 2^(n-1); T[n_, k_]:= T[n, k] = (k-1)*T[n-1, k] + (n-k+2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

PROG

(PARI) {T(n, k) = if(k==1, 1, if(k==n, 2^(n-1), (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)))};

for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

(Sage)

def T(n, k):

if (k==1):

return 1

elif (k==n):

return 2^(n-1)

else: return (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)

[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019

CROSSREFS

Cf. A000096 (column k=1), A002627, A008517.

Cf. This sequence (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3).

Sequence in context: A128718 A112358 A126351 * A246173 A092821 A238241

Adjacent sequences: A157008 A157009 A157010 * A157012 A157013 A157014

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 21 2009

STATUS

approved

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Last modified December 6 16:00 EST 2022. Contains 358644 sequences. (Running on oeis4.)