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A176200 A symmetrical triangle T(n, m) = 2*Eulerian(n+1, m) -1, read by rows. 2
1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 131, 51, 1, 1, 113, 603, 603, 113, 1, 1, 239, 2381, 4831, 2381, 239, 1, 1, 493, 8585, 31237, 31237, 8585, 493, 1, 1, 1003, 29215, 176467, 312379, 176467, 29215, 1003, 1, 1, 2025, 95679, 910383, 2620707, 2620707, 910383, 95679, 2025, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 9, 44, 235, 1434, 10073, 80632, 725751, 7257590, 79833589, ...}.

LINKS

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

FORMULA

T(n, m) = 2*Eulerian(n+1, m) - 1, where Eulerian(n, k) = A008292(n,k).

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,   7,    1;

  1,  21,   21,     1;

  1,  51,  131,    51,     1;

  1, 113,  603,   603,   113,    1;

  1, 239, 2381,  4831,  2381,  239,   1;

  1, 493, 8585, 31237, 31237, 8585, 493, 1;

MATHEMATICA

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];

T[n_, m_]:= 2*Eulerian[n+1, m]-1;

Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 25 2019 *)

PROG

(PARI) Eulerian(n, k) = sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k-j+1)^n); {T(n, k) = 2*Eulerian(n+1, k) - 1 };

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 25 2019

(MAGMA) Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;

[[2*Eulerian(n+1, k)-1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 25 2019

(Sage)

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

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

[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 25 2019

CROSSREFS

Cf. A008292, A109128.

Sequence in context: A174033 A119727 A157272 * A046739 A056752 A053714

Adjacent sequences:  A176197 A176198 A176199 * A176201 A176202 A176203

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Apr 11 2010

EXTENSIONS

Edited by G. C. Greubel, Apr 25 2019

STATUS

approved

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Last modified March 28 11:49 EDT 2020. Contains 333083 sequences. (Running on oeis4.)