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A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). 4
1, 1, -1, 1, -4, 1, 1, -15, 5, -1, 1, -58, 10, -6, 1, 1, -229, -66, -26, 7, -1, 1, -912, -1017, -288, 23, -8, 1, 1, -3643, -8733, -4779, -415, -41, 9, -1, 1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1, 1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums are: {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202,...}.

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=1..100 of triangle, flattned

FORMULA

e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

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

EXAMPLE

Triangle begins with:

1.

1,     -1.

1,     -4,       1.

1,    -15,       5,      -1.

1,    -58,      10,      -6,       1.

1,   -229,     -66,     -26,       7,     -1.

1,   -912,   -1017,    -288,      23,     -8,     1.

1,  -3643,   -8733,   -4779,    -415,    -41,     9,   -1.

1, -14566,  -61880,  -63606,  -17242,  -1158,    40,  -10,   1.

1, -58257, -396796, -691036, -375118, -60990, -1956,  -60,  11,  -1.

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_]:=(-1)^(n-1); T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] + (n-k-1)*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, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* 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 (-1)^(n-1)

    else: return (k+2)*T(n-1, k) + (n-k-1)* 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. A008517.

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), This Sequence (m=3).

Sequence in context: A058711 A202906 A177984 * A141724 A208956 A271705

Adjacent sequences:  A157010 A157011 A157012 * A157014 A157015 A157016

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 21 2009

STATUS

approved

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Last modified April 22 09:59 EDT 2021. Contains 343174 sequences. (Running on oeis4.)