|
| |
|
|
A101860
|
|
a(n) = (3+n)*(2 + 33*n + n^2)/6.
|
|
3
|
|
|
|
1, 24, 60, 110, 175, 256, 354, 470, 605, 760, 936, 1134, 1355, 1600, 1870, 2166, 2489, 2840, 3220, 3630, 4071, 4544, 5050, 5590, 6165, 6776, 7424, 8110, 8835, 9600, 10406, 11254, 12145, 13080, 14060, 15086, 16159, 17280, 18450, 19670, 20941
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
The 4th partial summation within series as series accumulate n times from an initial sequence of Euler Triangle's row 4: 1,11,11,1.
The partial sums of A101859 (plus a leading 1). 4th row in the array shown in the examples. The 2nd column is A101104, the 3rd column is A101103, the 4th column is A005914.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: ( 1 + 20*x - 30*x^2 + 10*x^3 ) / (x-1)^4 . - R. J. Mathar, Dec 06 2011
|
|
|
EXAMPLE
|
Array with first column equal to the 4th row of A008292, and column k defined by partial sums of the preceding column k-1:
1 1 1 1 1 1 1 1 1 1 1
11 12 13 14 15 16 17 18 19 20 21
11 23 36 50 65 81 98 116 135 155 176
1 24 60 110 175 256 354 470 605 760 936 A101860
0 24 84 194 369 625 979 1449 2054 2814 3750 A101861
0 24 108 302 671 1296 2275 3724 5778 8592 12342 A101862
0 24 132 434 1105 2401 4676 8400 14178 22770 35112
0 24 156 590 1695 4096 8772 17172 31350 54120 89232
0 24 180 770 2465 6561 15333 32505 63855 117975 207207
... ... ... ... ... ... ... ... ... ...
|
|
|
MATHEMATICA
|
LinearRecurrence[{4, -6, 4, -1}, {1, 24, 60, 110}, 50] (* or *) CoefficientList[Series[(1+20*x-30*x^2+10*x^3)/(x-1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *)
|
|
|
PROG
|
(Magma) I:=[1, 24, 60, 110]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 26 2012
(PARI) a(n) = (n+3)*(n^2+33*n+2)/6; \\ Altug Alkan, Sep 23 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 18 2004
|
|
|
STATUS
|
approved
|
| |
|
|
