A157012 - OEIS

A157012

Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 14, 1, 0, 1, 58, 110, 33, 1, 0, 1, 179, 672, 495, 72, 1, 0, 1, 543, 3583, 5163, 1917, 151, 1, 0, 1, 1636, 17590, 43730, 32154, 6808, 310, 1, 0, 1, 4916, 81812, 324190, 411574, 176272, 22904, 629, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Row sums are:

{1, 1, 2, 7, 34, 203, 1420, 11359, 102230, 1022299,...}.

This recursion set doesn't seem to produce the Eulerian 2nd A008517.

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

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

e(n,k,m) = (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) for m=2.

T(n,k) = (k+2)*T(n-1, k) + (n-k)*T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0. - G. C. Greubel, Feb 22 2019

EXAMPLE

Triangle begins with:

1, 0.

1, 1, 0.

1, 5, 1, 0.

1, 18, 14, 1, 0.

1, 58, 110, 33, 1, 0.

1, 179, 672, 495, 72, 1, 0.

1, 543, 3583, 5163, 1917, 151, 1, 0.

1, 1636, 17590, 43730, 32154, 6808, 310, 1, 0.

1, 4916, 81812, 324190, 411574, 176272, 22904, 629, 1, 0.

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_, 0]:= 1; T[n_, n_]:= 0; T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] +(n-k) *T[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

PROG

(PARI) {T(n, k) = if(k==0, 1, if(k==n, 0, (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)))};

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

(Sage)

def T(n, k):

if (k==0): return 1

elif (k==n): return 0

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

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 22 2019

CROSSREFS

Cf. A008517.

Cf. A157011 (m=0), A008292 (m=1), This Sequence (m=2), A157013 (m=3).