Original entry on oeis.org
0, 0, 1, 4, 19, 107, 682, 4915, 39871, 361138, 3621531, 39884367, 478847750, 6226248403, 87174202427, 1307651621142, 20922657286067, 355686620215179, 6402368573492818, 121645066483568099, 2432901775271051559, 51090940513901948778
Offset: 0
-
a = Table[n! - Sum[(
Eulerian[n - j, j]), {j, 0, Floor[n/2]}], {n, 0, 30}]
A178118
Antidiagonal sums of the triangle A060187.
Original entry on oeis.org
0, 1, 1, 2, 7, 25, 100, 469, 2481, 14406, 90995, 621553, 4561112, 35736921, 297435521, 2618575194, 24297706927, 236870849417, 2419213831452, 25820011544781, 287327296473585, 3326999636488190, 40011485288491131
Offset: 0
- David M. Burton, Elementary number theory, McGraw Hill (2002), page 286
-
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]]
a[n_] := Sum[f[n - m - 1, m], {m, 0, Floor[(n - 1)/2]}]
Table[a[n], {n, 0, 30}]
Exact definition moved to formula - the Assoc. Eds. of the OEIS, Aug 20 2010
A178134
Sum_{m=0..(n-1)/2} A176263(n-m-1, m).
Original entry on oeis.org
0, 1, 1, 2, -3, -2, -32, -81, -311, -810, -2515, -6864, -19944, -55043, -156023, -433522, -1217427, -3391226, -9488456, -26462205, -73933535, -206293134, -576040339, -1607642688, -4488069168, -12526662167, -34967630447
Offset: 0
-
A178134 := proc(n)
add( A176263(n-m-1,m), m=0..(n-1)/2) ;
end proc: # R. J. Mathar, May 15 2016
-
Clear[a, f, a0, t]
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
t[n_, m_, a_] := f[m + 1, a] + f[n + 1 - m, a] - f[n + 1, a];
a = 5;
a0[n_] := Sum[t[n - m - 1, m, a], {m, 0, Floor[(n - 1)/2]}];
Table[a0[n], {n, 0, 30}]
-
a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 25,5,-25,-4,-6,-2,7,1]^n*[0;1;1;2;-3;-2;-32;-81])[1,1] \\ Charles R Greathouse IV, May 15 2016
A174958
a(n)=Sum((A008292(n - j, j) - C(n - j - 1, j))/2, j=0, [(n - 1)/2]).
Original entry on oeis.org
0, 0, 0, 0, 1, 4, 15, 56, 214, 854, 3607, 16172, 76853, 386082, 2044198, 11373124, 66300473, 403939612, 2566116299, 16962629860, 116452790838, 828903740138, 6107712000563, 46521422681724, 365811331693305, 2965957618809246, 24767913121016790, 212803409969904264
Offset: 0
- Burton, David M.,Elementary number theory,McGraw Hill,N.Y.,2002,p 286, problem 23
-
a = Table[Sum[(Eulerian[n -
j, j] - Binomial[n - j - 1, j])/2, {j, 0,
Floor[(n - 1)/2]}], {n, 0, 30}]
A174993
a(n) = -Floor[n/2]! + Sum[(Eulerian[n - j, j]), {j, 0, Floor[n/2]}].
Original entry on oeis.org
0, 0, 1, 4, 11, 36, 119, 443, 1718, 7245, 32313, 153730, 771677, 4088053, 22741818, 132596893, 807840501, 5132194862, 33924901021, 232905225561, 1657803862422, 12215420390037, 93042805475305, 731622623516178, 5931914758691917, 49535825763153373, 425606813712984146, 3758735172560999933, 34089943206777076429, 317245175458777517030
Offset: 1
A344393
T(n, k) = Eulerian1(n - k, k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 11, 1, 1, 26, 11, 0, 1, 57, 66, 1, 1, 120, 302, 26, 0, 1, 247, 1191, 302, 1, 1, 502, 4293, 2416, 57, 0, 1, 1013, 14608, 15619, 1191, 1, 1, 2036, 47840, 88234, 15619, 120, 0, 1, 4083, 152637, 455192, 156190, 4293, 1
Offset: 0
Triangle starts:
[ 0] [1]
[ 1] [1]
[ 2] [1, 0]
[ 3] [1, 1]
[ 4] [1, 4, 0]
[ 5] [1, 11, 1]
[ 6] [1, 26, 11, 0]
[ 7] [1, 57, 66, 1]
[ 8] [1, 120, 302, 26, 0]
[ 9] [1, 247, 1191, 302, 1]
[10] [1, 502, 4293, 2416, 57, 0]
[11] [1, 1013, 14608, 15619, 1191, 1]
-
T := (n, k) -> combinat:-eulerian1(n - k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);
Showing 1-6 of 6 results.
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