A156363 - OEIS

%I #12 Sep 08 2022 08:45:41

%S 1,1,3,1,6,25,1,13,76,350,1,28,242,1430,6951,1,59,783,6023,35659,

%T 179487,1,122,2527,25782,187092,1108128,5715424,1,249,8070,110960,

%U 995595,6963711,41250694,216627840,1,504,25456,476626,5337322,44302510,302087532,1789534102,9528822303

%N A triangle sequence related to the Eulerian numbers of the second kind: t(n,m) = Sum_{i=0..m}(-1)^(m-i)*binomial(n-i-1, m-i)*Stirling2(n+i+1, i+1).

%C Row sums are: {1, 4, 32, 440, 8652, 222012, 7039076, 265957120, 11670586356, 583472429540, 32744436653656,...}

%H G. C. Greubel, <a href="/A156363/b156363.txt">Rows n=0..100 of triangle, flattened</a>

%H L.  Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/BSfront.html">Completion of a Rational Function Sequence of Carlitz</a>, page 3.

%F t(n,m) = Sum_{i=0..m}(-1)^(m-i)*binomial(n-i-1, m-i)*Stirling2(n+i+1, i+1).

%e Triangle begins as:

%e 1;

%e 1,   3;

%e 1,   6,    25;

%e 1,  13,    76,    350;

%e 1,  28,   242,   1430,   6951;

%e 1,  59,   783,   6023,  35659,  179487;

%e 1, 122,  2527,  25782, 187092, 1108128,  5715424;

%e 1, 249,  8070, 110960, 995595, 6963711, 41250694, 216627840;

%t t[n_, m_] = Sum[(-1)^(m-i)*Binomial[n-i-1, m-i]*StirlingS2[n+i+1, i+1], {i, 0, m}]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]//Flatten

%o (PARI) {t(n,m) = sum(j=0,m, (-1)^(m-j)*binomial(n-j-1, m-j)*stirling(n+j +1, j+1,2))};

%o for(n=0,10, for(m=0,n, print1(t(n,m), ", "))) \\ _G. C. Greubel_, Feb 24 2019

%o (Magma) [[(&+[(-1)^(m-j)*Binomial(n-j-1, m-j)*StirlingSecond(n+j+1, j+1): j in [0..m]]): m in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Feb 24 2019

%o (Sage) [[sum((-1)^(m-j)*binomial(n-j-1, m-j)*stirling_number2(n+j+1, j+1) for j in (0..m)) for m in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 24 2019

%Y Cf. A048993 (Stirling2), A008277, A156139, A156364.

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_, Feb 08 2009