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.

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

Views

Author

Paul D. Hanna, Dec 15 2018

Keywords

Examples

			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, ...].

Links

Crossrefs

Cf. A322236, A322237, A322238.
Cf. A322225 (variant), A322891 (variant).

Programs

  • Mathematica
    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 *)
  • 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(""))

Formula

Row sums equal (2*n+1)!/(n!*2^n), the odd double factorials.
Left and right borders equal n!.

A322236 a(n) = A322237(n) / (n*(n+1)/2), where A322237(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.

Original entry on oeis.org

1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, 11181850967, 266546940947, 6069884741155, 169005305069771, 4510734458734443, 143664066858425883, 4399531515393236907, 157747037226275555718, 5453223770914252146978, 217372015577641986139848, 8374038291341888594002908, 367340884744321785348071011, 15606634300050239405862650475
Offset: 1

Views

Author

Paul D. Hanna, Dec 15 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 A322238.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal (A322237), beginning
[1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
may be divided by triangular numbers to obtain this sequence
[1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].

Links

Crossrefs

Cf. A322237, A322238.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}]/ (n(n+1)/2);
Array[a, 24] (* Jean-François Alcover, Dec 28 2018 *)
  • 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=1, 30, print1( T(n, n-1)/(n*(n+1)/2), ", "))

A322237 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.

Original entry on oeis.org

1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, 39093550782, 872184375426, 24255771626177, 637337897821275, 20280636608372520, 613459886387884248, 21980602229339160099, 752319889132243511097, 29971937072992355586420, 1145176991891992950865380, 50212935598435298798304888, 2118631687709497814282735724, 101386084189432812756067599036
Offset: 1

Views

Author

Paul D. Hanna, Dec 15 2018

Keywords

Comments

a(n) = n*(n+1)/2 * A322236(n) for n >= 1.

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 this sequence forms a diagonal.

Links

Crossrefs

Cf. A322235, A322238, A322236.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}];
Array[a, 23] (* Jean-François Alcover, Dec 28 2018 *)
  • 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=1,30, print1( T(n,n-1),", "))

A322228 a(n) = [x^n] Product_{k=1..n} (k + x - k*x^2), for n >= 0.

Original entry on oeis.org

1, 1, -3, -21, 75, 1475, -5005, -221389, 593523, 57764619, -89101881, -23273632371, 953636541, 13409519997705, 23908442020749, -10469975115603501, -40844292735050541, 10646036726696597027, 66995992524016223543, -13672657170891872702719, -122282221141986787179519, 21647316686778755963070321, 256325163531592225309743129, -41426918732532942751217361155, -620418821801458605268716606275, 94275566307675915918535250768725
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2018

Keywords

Comments

a(n+1) = -2*(n+1) * A322227(n) + a(n), for n >= 1.
a(n+1) = -n*(n+1)^2 * A322226(n) + a(n), for n >= 1.

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 this sequence.
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 A322226:
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].

Links

Crossrefs

Cf. A322235, A322236, A322237.
Cf. A322238 (variant).

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n}];
Array[a, 26, 0] (* Jean-François Alcover, Dec 29 2018 *)
  • 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), ", "))

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.
Showing 1-5 of 5 results.

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

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