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Triangle read by rows: T(n,m) = (-1)^n*Sum_{i=0..m} [(-1)^(m-i)*binomial(n- i-1, m-i)*Stirling_1(n+i+1,i+1), for 0 <= m <= n.
1

%I #9 May 01 2013 21:09:56

%S 1,1,3,2,9,35,6,38,181,735,24,202,1148,5395,22449,120,1284,8560,45832,

%T 213529,902055,720,9468,73052,440516,2275270,10540509,44990231,5040,

%U 79344,700380,4713740,26783190,135134070,623527389,2681453775,40320

%N Triangle read by rows: T(n,m) = (-1)^n*Sum_{i=0..m} [(-1)^(m-i)*binomial(n- i-1, m-i)*Stirling_1(n+i+1,i+1), for 0 <= m <= n.

%C Row sums are {1, 4, 46, 960, 29218, 1171380, 58329766, 3472396928, 240584307106, 19018858710852, 1689457066042590, ...}. - see A191870.

%e Triangle begins

%e {1},

%e {1, 3},

%e {2, 9, 35},

%e {6, 38, 181, 735},

%e {24, 202, 1148, 5395, 22449},

%e {120, 1284, 8560, 45832, 213529, 902055},

%e {720, 9468, 73052, 440516, 2275270, 10540509, 44990231},

%e {5040, 79344, 700380, 4713740, 26783190, 135134070, 623527389, 2681453775},

%e {40320, 744336, 7440840, 55477636, 344589972, 1883062894, 9345564224, 42994209331, 185953177553},

%e {362880, 7725600, 86678136, 711387192, 4804798156, 28306022216, 150469912896, 737690211448, 3386028203405, 14710753408923},

%e {3628800, 87884640, 1097845632, 9863244552, 72108611456, 456146630800, 2587934993000, 13476479586304, 65448246877514, 299870086199497, 1307535010540395}

%e For example, T(4,2) = 1148 = C(3,2)S1(5,1)-C(2,1)S1(6,2)+C(1,0)S1(7,3).

%t t[n_, m_] = (-1)^n*Sum[(-1)^(m - i)*Binomial[n - i - 1, m - i]*StirlingS1[n + i + 1, i + 1], {i, 0, m}];

%t Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];

%t Flatten[%]

%Y Cf. A191870.

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_, Feb 09 2009

%E Corrected and edited by _N. J. A. Sloane_, Jun 18 2011