A322235
Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + k*x^2), for n >= 0.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 5, 3, 2, 6, 11, 24, 23, 24, 11, 6, 24, 50, 131, 160, 215, 160, 131, 50, 24, 120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120, 720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720, 5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040, 40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320, 362880, 1026576, 4438620, 8936288, 21541905, 33149481, 56464695, 68597418, 89489025, 86715299, 89489025, 68597418, 56464695, 33149481, 21541905, 8936288, 4438620, 1026576, 362880
Offset: 0
This irregular triangle formed from coefficients of x^k in Product_{m=1..n} (m + x + m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 1;
2, 3, 5, 3, 2;
6, 11, 24, 23, 24, 11, 6;
24, 50, 131, 160, 215, 160, 131, 50, 24;
120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
in which the central terms equal A322238.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322237 in the above triangle
[1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
may be divided by triangular numbers to obtain A322236:
[1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].
-
row[n_] := CoefficientList[Product[k+x+k*x^2, {k, 1, n}] + O[x]^(2n+1), x];
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 26 2018 *)
-
{T(n, k) = polcoeff( prod(m=1, n, m + x + 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(""))
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), ", "))
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), ", "))
A322893
a(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.
Original entry on oeis.org
1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, 23818189395, 907365113622, 24884202594186, 1097379059482797, 35843982129214455, 1794829778206820280, 68106808437178597960, 3815489686616468849025, 165072679883587905823683, 10226191400763164277215330, 497092886801366317217274750, 33732223801436694239674078341, 1820835126778068312737993859263
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 this sequence forms a secondary diagonal in the above triangle
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...]
and 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=1, 30, print1( A322891(n, n-1), ", "))
Showing 1-4 of 4 results.