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A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i). 3
1, 1, 0, 3, 2, 2, 13, 10, 8, 6, 75, 62, 52, 44, 38, 541, 466, 404, 352, 308, 270, 4683, 4142, 3676, 3272, 2920, 2612, 2342, 47293, 42610, 38468, 34792, 31520, 28600, 25988, 23646, 545835, 498542, 455932, 417464, 382672, 351152, 322552, 296564, 272918 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

FORMULA

Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).

EXAMPLE

Triangle begins as:

     1;

     1,    0;

     3,    2,    2;

    13,   10,    8,    6;

    75,   62,   52,   44,   38;

   541,  466,  404,  352,  308,  270;

  4683, 4142, 3676, 3272, 2920, 2612, 2342;

  ...

MAPLE

T:= (n, k)-> k!*(n-k)!*coeff(series(coeff(series(exp(-y)/

        (2-exp(x+y)), y, k+1), y, k), x, n-k+1), x, n-k):

seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 02 2019

# second Maple program:

b:= proc(n) option remember; `if`(n<2, 1,

      add(b(n-j)*binomial(n, j), j=1..n))

    end:

T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n-j), j=0..k):

seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 02 2019

MATHEMATICA

A000670[n_]:= If[n==0, 1, Sum[k! StirlingS2[n, k], {k, n}]]; T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*A000670[n-k+j], {j, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)

PROG

(PARI)

A000670(n) = sum(k=0, n, k!*stirling(n, k, 2));

T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(k, j)*A000670(n-k+j));

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 02 2019

(MAGMA)

A000670:= func< n | &+[Factorial(k)*StirlingSecond(n, k): k in [0..n]] >;

[(&+[(-1)^(k-j)*Binomial(k, j)*A000670(n-k+j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019

(Sage)

def A000670(n): return sum(factorial(k)*stirling_number2(n, k) for k in (0..n))

def T(n, k): return sum((-1)^(k-j)*binomial(k, j)*A000670(n-k+j) for j in (0..k))

[[T(n, k) for k in (0..n)] for n in (0..10)]

(GAP)

A000670:= function(n)

     return Sum([0..n], i-> Factorial(i)*Stirling2(n, i) ); end;

T:= function(n, k)

    return Sum([0..k], j-> (-1)^(k-j)*Binomial(k, j)*A000670(n-k+j) ); end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Oct 02 2019

CROSSREFS

Columns k=0-1 give: A000670, A232472.

Row sums give A089677(n+1).

Main diagonal gives A052841.

Cf. A005649, A069321, A073146.

Sequence in context: A074248 A266004 A206703 * A108032 A053370 A016458

Adjacent sequences:  A122098 A122099 A122100 * A122102 A122103 A122104

KEYWORD

easy,nonn,tabl

AUTHOR

Vladeta Jovovic, Oct 18 2006

STATUS

approved

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)