Original entry on oeis.org
0, 64, 784, 4368, 16368, 48048, 119392, 262752, 527136, 983136, 1728496, 2894320, 4651920, 7220304, 10874304, 15953344, 22870848, 32124288, 44305872, 60113872, 80364592, 106004976, 138125856, 177975840, 226975840, 286734240, 359062704, 445992624, 549792208, 672984208, 818364288, 989020032, 1188350592, 1420086976, 1688312976, 1997486736, 2352462960, 2758515760, 3221362144, 3747186144
Offset: 1
A001823
Central factorial numbers: column 2 in triangle A008956.
Original entry on oeis.org
0, 9, 259, 1974, 8778, 28743, 77077, 179452, 375972, 725781, 1312311, 2249170, 3686670, 5818995, 8892009, 13211704, 19153288, 27170913, 37808043, 51708462, 69627922, 92446431, 121181181, 157000116, 201236140, 255401965, 321205599, 400566474, 495632214
Offset: 1
- J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 1..1000
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
-
A001823:=-(9+196*z+350*z**2+84*z**3+z**4)/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
-
Table[1/90*n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n + 7), {n, 40}] (* Stefan Steinerberger, Apr 15 2006 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,9,259,1974,8778,28743,77077},30] (* Harvey P. Dale, Jun 09 2013 *)
A182971
Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.
Original entry on oeis.org
1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
Offset: 0
Triangle begins:
1
1, -1
1, -4
1, -10, 9
1, -20, 64
1, -35, 259, -225
1, -56, 784, -2304
1, -84, 1974, -12916, 11025
1, -120, 4368, -52480, 147456
1, -165, 8778, -172810, 1057221, -893025
1, -220, 16368, -489280, 5395456, -14745600
...
E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.
-
Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 12 do
t1:=eval(Q(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:
A108684
a(n) = (n+1)*(n+2)*(n+3)*(19*n^3 + 111*n^2 + 200*n + 120)/720.
Original entry on oeis.org
1, 15, 93, 372, 1141, 2926, 6594, 13476, 25509, 45397, 76791, 124488, 194649, 295036, 435268, 627096, 884697, 1224987, 1667953, 2237004, 2959341, 3866346, 4993990, 6383260, 8080605, 10138401, 12615435, 15577408, 19097457, 23256696
Offset: 0
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 10).
-
a:=n->(n+1)*(n+2)*(n+3)*(19*n^3+111*n^2+200*n+120)/720: seq(a(n),n=0..33);
-
Table[(n + 1) (n + 2) (n + 3) (19 n^3 + 111 n^2 + 200 n + 120)/720, {n, 0, 29}] (* or *)
CoefficientList[Series[(1 + 8 x + 9 x^2 + x^3)/(1 - x)^7, {x, 0, 29}], x] (* or *)
Table[Sum[Binomial[(n + 1 - k) + 1, 2] Apply[Subtract, Map[Binomial[# + 2, 3] &, {n + 1, k}]], {k, 0, n}], {n, 0, 29}] (* Michael De Vlieger, Jun 08 2017 *)
Showing 1-4 of 4 results.
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