A257602
Expansion of (1 + x + 21*x^2 + x^3 + x^4)/(1 - x)^5.
Original entry on oeis.org
1, 6, 41, 156, 426, 951, 1856, 3291, 5431, 8476, 12651, 18206, 25416, 34581, 46026, 60101, 77181, 97666, 121981, 150576, 183926, 222531, 266916, 317631, 375251, 440376, 513631, 595666, 687156, 788801, 901326, 1025481, 1162041, 1311806, 1475601, 1654276, 1848706, 2059791, 2288456, 2535651
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
-
I:=[1,6,41,156,426]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Jun 08 2015
-
CoefficientList[Series[(1 +x +21x^2 +x^3 +x^4)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 08 2015 *)
LinearRecurrence[{5,-10,10,-5,1},{1,6,41,156,426},40] (* Harvey P. Dale, Dec 01 2017 *)
-
[1 + 5*n*(n+1)*(5*n^2+5*n+2)/24 for n in (0..50)] # G. C. Greubel, Mar 24 2022
A373423
Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(0) = 1, cf(1) = -1/(x - 1), and for n > 1 is cf(n) = ~( ~x - 1/(~x - 1/(~x - 1/(~x - 1/(~x - ... 1/(~x + 1))))...) ) where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 5, 6, 5, 1, 1, 0, 1, 6, 10, 14, 8, 1, 1, 0, 1, 7, 15, 30, 31, 13, 1, 1, 0, 1, 8, 21, 55, 85, 70, 21, 1, 1, 0, 1, 9, 28, 91, 190, 246, 157, 34, 1, 1, 0, 1, 10, 36, 140, 371, 671, 707, 353, 55, 1, 1, 0
Offset: 0
Generating functions of row n:
gf0 = 1;
gf1 = - 1/( x-1);
gf2 = x + 1/(-x+1);
gf3 = x - 1/( x-1/( x+1));
gf4 = x + 1/(-x-1/(-x-1/(-x+1)));
gf5 = x - 1/( x-1/( x-1/( x-1/( x+1))));
gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1)))));
.
Array begins:
[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
[1] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[2] 1, 2, 1, 1, 1, 1, 1, 1, 1, ... A373565
[3] 1, 3, 3, 5, 8, 13, 21, 34, 55, ... A373566
[4] 1, 4, 6, 14, 31, 70, 157, 353, 793, ... A373567
[5] 1, 5, 10, 30, 85, 246, 707, 2037, 5864, ... A373568
[6] 1, 6, 15, 55, 190, 671, 2353, 8272, 29056, ... A373569
A000217, A006322, A108675, ...
A000330, A085461, A244881, ...
.
Triangle starts:
[0] 1;
[1] 1, 0;
[2] 1, 1, 0;
[3] 1, 2, 1, 0;
[4] 1, 3, 1, 1, 0;
[5] 1, 4, 3, 1, 1, 0;
[6] 1, 5, 6, 5, 1, 1, 0;
-
row := proc(n, len) local x, a, j, ser;
if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then
a := x + 1; for j from 1 to n-1 do a := x - 1 / a od: else
a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi;
ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]: # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
seq(lprint([n], row(n, 9)), n = 0..9);
-
def Arow(n, len):
R. = PowerSeriesRing(ZZ, len)
if n == 0: return [1] + [0]*(len - 1)
if n == 1: return [1]*(len - 1)
x = x if n % 2 == 1 else -x
a = x + 1
for _ in range(n - 1):
a = x - 1 / a
if n % 2 == 0: a = -a
return a.list()
for n in range(8): print(Arow(n, 9))
Comments