A322894
a(n) = A322893(n) / (n*(n+1)/2), where A322893(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.
Original entry on oeis.org
1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, 13747956267, 319028238387, 12059110543767, 341371258373471, 14956914818390169, 500785356155724985, 24937841088996528425, 965337309260747987273, 53822060004016654090607, 2367108984768411034367975, 146026942863362312725861811, 7196976785684064477225272171, 486563915009872154819986680357
Offset: 1
The irregular triangle A322891 of coefficients of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 2;
2, 3, 9, 6, 8;
6, 11, 42, 45, 84, 44, 48;
24, 50, 227, 310, 717, 620, 908, 400, 384;
120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
Note that the terms in the secondary diagonal A322893 in the above triangle,
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...],
may be divided by triangular numbers n*(n+1)/2 to obtain this sequence:
[1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
-
{A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
/* Print this sequence */
for(n=1, 30, print1( A322891(n, n-1)/(n*(n+1)/2), ", "))
A322891
Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 9, 6, 8, 6, 11, 42, 45, 84, 44, 48, 24, 50, 227, 310, 717, 620, 908, 400, 384, 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840, 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080, 5040, 13068, 83692, 163585, 556640, 838554, 1948268, 2254625, 3896536, 3354216, 4453120, 2617360, 2678144, 836352, 645120, 40320, 109584, 763244, 1601460, 5955777, 9882432, 25330938, 33402132, 64599201, 66804264, 101323752, 79059456, 95292432, 51246720, 48847616, 14026752, 10321920
Offset: 0
This irregular triangle of coefficients T(n,k) of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 2;
2, 3, 9, 6, 8;
6, 11, 42, 45, 84, 44, 48;
24, 50, 227, 310, 717, 620, 908, 400, 384;
120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080;
5040, 13068, 83692, 163585, 556640, 838554, 1948268, 2254625, 3896536, 3354216, 4453120, 2617360, 2678144, 836352, 645120;
40320, 109584, 763244, 1601460, 5955777, 9882432, 25330938, 33402132, 64599201, 66804264, 101323752, 79059456, 95292432, 51246720, 48847616, 14026752, 10321920; ...
in which the central terms equal A322892:
[1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, ...].
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322893 in this triangle,
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...],
may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
[1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
-
{T(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
A322892
a(n) = [x^n] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.
Original entry on oeis.org
1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, 44097148953, 1092097482333, 44645622936189, 1338624157833861, 62791851488870493, 2213430779241737793, 117082536584478235713, 4748345510312622896993, 279463602946698380026793, 12824987274099379222626701, 830920299335152521399853101, 42586722790649923167650932101, 3011022417317079016258969826109, 170527854080899363788154404878305
Offset: 0
The irregular triangle A322891 of coefficients of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 2;
2, 3, 9, 6, 8;
6, 11, 42, 45, 84, 44, 48;
24, 50, 227, 310, 717, 620, 908, 400, 384;
120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
in which the main diagonal forms this sequence.
Note that the terms in the secondary diagonal A322893 in the above triangle
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...]
may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
[1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
-
{A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
/* Print this sequence */
for(n=0, 30, print1( A322891(n, n), ", "))
Showing 1-3 of 3 results.
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