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!.

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.

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

Original entry on oeis.org

1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, -1009469538, -515714090814, -374961500823, 349796118587475, 949197425607720, -314320029983283752, -1565276549925545181, 361569820089891813849, 2715239099277372861920, -518323783521922446434520, -5333587428291215212424382, 906157476001402934272328354, 12062331313935951302447900940, -1897919702589547490476079347500, -31441371048822199544956413616625
Offset: 1

Views

Author

Paul D. Hanna, Dec 15 2018

Keywords

Comments

a(n) = n*(n+1)/2 * A322226(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 this sequence forms a diagonal.
RELATED SEQUENCES.
Note that the terms in this sequence
[1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
may be divided by triangular numbers n*(n+1)/2 to obtain A322226:
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].

Links

Crossrefs

Cf. A322225, A322226.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n-1}];
Array[a, 25] (* 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=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), ", "))
Showing 1-5 of 5 results.

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