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 1-4 of 4 results.

A322225 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, -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, 40320, 109584, -204436, -699804, 442665, 1969380, -575310, -3176172, 593523, 3176172, -575310, -1969380, 442665, 699804, -204436, -109584, 40320, 362880, 1026576, -2093220, -7488928, 5124105, 24465321, -7192395, -46885278, 7343325, 57764619, -7343325, -46885278, 7192395, 24465321, -5124105, -7488928, 2093220, 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, -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;
40320, 109584, -204436, -699804, 442665, 1969380, -575310, -3176172, 593523, 3176172, -575310, -1969380, 442665, 699804, -204436, -109584, 40320; ...
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 A322226:
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].

Links

Crossrefs

Cf. A322226, A322227, A322228.
Cf. A322235 (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

Each row sums to 1.
Left and right borders equal n! and (-1)^n*n!, respectively.

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

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 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, ...].

Links

Crossrefs

Cf. A322891, A322892, A322893.
Cf. A322226 (variant), A322236 (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)/(n*(n+1)/2), ", "))

Formula

a(n) = A322891(n, n-1) / (n*(n+1)/2).
a(n) = A322891(n, n+1) / (n*(n+1)).
a(n) appears to be odd 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 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.