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
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, ...].
-
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 *)
-
{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), ", "))
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
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, ...].
-
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(""))
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
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, ...].
-
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 *)
-
{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 1-3 of 3 results.
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