cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A322238 a(n) = [x^n] Product_{k=1..n} (k + x + k*x^2), for n >= 0.

Original entry on oeis.org

1, 1, 5, 23, 215, 1815, 24495, 302619, 5320667, 86715299, 1876495799, 38014052089, 976259270857, 23653053031933, 702814658564889, 19822951593203139, 668803323061123779, 21526439460249188211, 812828119716458951775, 29400983906741712373461, 1228278466826435935830261, 49325712126290139872176221, 2258694878457443286997591293, 99715752513094342744003434597, 4966247793605869355035248188325
Offset: 0

Views

Author

Paul D. Hanna, Dec 12 2018

Keywords

Examples

			The irregular triangle A322235 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 this sequence.
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, ...].

Links

Crossrefs

cF. A322235, A322236, A322237.
Cf. A322228 (variant), A322894 (variant).

Programs

  • PARI
    {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(""))
/* Print this sequence */
for(n=0,30, print1( T(n,n),", "))

Formula

a(n+1) = 2*(n+1) * A322237(n) + a(n), for n >= 1.
a(n+1) = n*(n+1)^2 * A322236(n) + a(n), for n >= 1.

A322226 a(n) = A322227(n) / (n*(n+1)/2), where A322227(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.

Original entry on oeis.org

1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, -6611719113, -4120456053, 3331391605595, 7909978546731, -2311176691053557, -10230565685787877, 2114443392338548619, 14290732101459857168, -2468208492961535459212, -23089123066195736850322, 3581650102772343613724618, 43704098963536055443651815, -6326399008631824968253597825, -96742680150222152446019734205
Offset: 1

Views

Author

Paul D. Hanna, Dec 15 2018

Keywords

Examples

			The irregular triangle A322225 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, -3, -3, 2;
6, 11, -12, -21, 12, 11, -6;
24, 50, -61, -140, 75, 140, -61, -50, 24;
120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120;
720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720;
5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040; ...
in which the central terms equal A322228.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322227 in the above triangle
[1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
may be divided by triangular numbers to obtain this sequence
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].

Links

Crossrefs

Cf. A322227, A322228.
Cf. A322236 (variant), A322894 (variant).

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n - 1}]/(n (n + 1)/2);
Array[a, 25] (* Jean-François Alcover, Dec 29 2018 *)
  • PARI
    {A322225(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( A322225(n, k), ", ")); print(""))
/* Print this sequence */
for(n=1, 30, print1( A322225(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

Views

Author

Paul D. Hanna, Dec 29 2018

Keywords

Comments

Row sums equal A007559(n+1), the triple factorial numbers.

Examples

			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, ...].

Links

Crossrefs

Cf. A322892 (central terms), A322893 (diagonal), A322894.
Cf. A322235 (variant), A322225 (variant), A000165, A007559.

Programs

  • PARI
    {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(""))

Formula

T(n,0) = n! for n >= 0.
T(n,2*n) = 2^n * n!, the even double factorials, for n >= 0.
Sum_{k=0..2*n} T(n,k) = Product_{k=0..n} (3*k + 1), the triple factorials, for n >= 0.

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

Views

Author

Paul D. Hanna, Dec 29 2018

Keywords

Examples

			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, ...].

Links

Crossrefs

Cf. A322891, A322893, A322894.
Cf. A322238 (variant).

Programs

  • PARI
    {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), ", "))

Formula

a(n+1) = 4*(n+1) * A322893(n) + a(n), for n >= 1.
a(n+1) = 2*n*(n+1)^2 * A322894(n) + a(n), for n >= 1.

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

Views

Author

Paul D. Hanna, Dec 29 2018

Keywords

Examples

			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, ...].

Links

Crossrefs

Cf. A322891, A322892, A322894.
Cf. A322237 (variant).

Programs

  • PARI
    {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), ", "))

Formula

a(n) = A322891(n, n-1) for n >= 1.
a(n) = A322891(n, n+1)/2 for n >= 1.
a(n) = n*(n+1)/2 * A322894(n) for n >= 1.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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