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.

A323224 A(n, k) = [x^k] C^n*x/(1 - x) where C = 2/(1 + sqrt(1 - 4*x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 9, 1, 0, 1, 5, 13, 22, 23, 1, 0, 1, 6, 19, 41, 64, 65, 1, 0, 1, 7, 26, 67, 131, 196, 197, 1, 0, 1, 8, 34, 101, 232, 428, 625, 626, 1, 0, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1
Offset: 0

Views

Author

Peter Luschny, Jan 24 2019

Keywords

Comments

Equals A096465 when the leading column (k = 0) is removed. - Georg Fischer, Jul 26 2023

Examples

			The square array starts:
   [n\k]  0  1   2   3    4     5     6      7       8       9
    ---------------------------------------------------------------
    [0]   0, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A057427
    [1]   0, 1,  2,  4,   9,   23,   65,   197,    626,   2056, ... A014137
    [2]   0, 1,  3,  8,  22,   64,  196,   625,   2055,   6917, ... A014138
    [3]   0, 1,  4, 13,  41,  131,  428,  1429,   4861,  16795, ... A001453
    [4]   0, 1,  5, 19,  67,  232,  804,  2806,   9878,  35072, ... A114277
    [5]   0, 1,  6, 26, 101,  376, 1377,  5017,  18277,  66727, ... A143955
    [6]   0, 1,  7, 34, 144,  573, 2211,  8399,  31655, 118865, ...
    [7]   0, 1,  8, 43, 197,  834, 3382, 13378,  52138, 201364, ...
    [8]   0, 1,  9, 53, 261, 1171, 4979, 20483,  82499, 327656, ...
    [9]   0, 1, 10, 64, 337, 1597, 7105, 30361, 126292, 515659, ...
.
Triangle given by ascending antidiagonals:
    0;
    0, 1;
    0, 1, 1;
    0, 1, 2,  1;
    0, 1, 3,  4,   1;
    0, 1, 4,  8,   9,   1;
    0, 1, 5, 13,  22,  23,   1;
    0, 1, 6, 19,  41,  64,  65,   1;
    0, 1, 7, 26,  67, 131, 196, 197,   1;
    0, 1, 8, 34, 101, 232, 428, 625, 626, 1;
.
The difference table of a column successively gives the preceding columns, here starting with column 6.
col(6) = 1, 65, 196, 428, 804, 1377, 2211, 3382, 4979, 7105, ...
col(5) =    64, 131, 232, 376,  573,  834, 1171, 1597, 2126, ...
col(4) =         67, 101, 144,  197,  261,  337,  426,  529, ...
col(3) =              34,  43,   53,   64,   76,   89,  103, ...
col(2) =                    9,   10,   11,   12,   13,   14, ...
col(1) =                          1,    1,    1,    1,    1, ...
col(0) =                                0,    0,    0,    0, ...
.
Example for the sum formula: C(0) = 1, C(1) = 1, C(2) = 2 and C(3) = 5.
X(3, 4) = {{0,0,0}, {0,0,1}, {0,1,0}, {1,0,0}, {0,0,2}, {0,1,1}, {0,2,0}, {1,0,1},
{1,1,0}, {2,0,0}, {0,0,3}, {0,1,2}, {0,2,1}, {0,3,0}, {1,0,2}, {1,1,1}, {1,2,0},
{2,0,1}, {2,1,0}, {3,0,0}}. T(3,4) = 1+1+1+1+2+1+2+1+1+2+5+2+2+5+2+1+2+2+2+5 = 41.

Crossrefs

The coefficients of the polynomials generating the columns are in A323233.
Sums of antidiagonals and row 1 are A014137. Main diagonal is A242798.
Rows: A057427 (n=0), A014137 (n=1), A014138 (n=2), A001453 (n=3), A114277 (n=4), A143955 (n=5).
Columns: A000027 (k=2), A034856 (k=3), A323221 (k=4), A323220 (k=5).
Similar array based on central binomials is A323222.
Cf. A096465.

Programs

  • Maple
    Row := proc(n, len) local C, ogf, ser; C := (1-sqrt(1-4*x))/(2*x);
ogf := C^n*x/(1-x); ser := series(ogf, x, (n+1)*len+1);
seq(coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do Row(n, 9) od;
# Alternatively by recurrence:
B := proc(n, k) option remember; if n <= 0 or k < 0 then 0
elif n = k then 1 else B(n-1, k) + B(n, k-1) fi end:
A := (n, k) -> B(n + k, k): seq(lprint(seq(A(n, k), k=0..9)), n=0..9);
  • Mathematica
    (* Illustrating the sum formula, not efficient. *) T[0, K_] := Boole[K != 0];
T[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
Sum[Product[CatalanNumber[m[[i]]], {i, 1, N}], {m , X[K]}]];
Trow[n_] := Table[T[n, k], {k, 0, 9}]; Table[Trow[n], {n, 0, 9}]

Formula

For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let C(m) denote the m-th Catalan number. Then: A(n, k) = Sum_{(j1,...,jn) in X(n, k)} C(j1)*C(j2)*...*C(jn).
A(n, k) = T(n + k, k) with T(n, k) = T(n-1, k) + T(n, k-1) with T(n, k) = 0 if n <= 0 or k < 0 and T(n, n) = 1.

A277956 a(n) = (n+2)*Sum_{i=0..n}(binomial(3*n-2*i+1, n-i)/(2*n-i+2)).

Original entry on oeis.org

1, 4, 19, 101, 573, 3382, 20483, 126292, 788878, 4976489, 31635811, 202354517, 1300880374, 8398175713, 54409200963, 353571026085, 2303666554659, 15043760670031, 98439176169692, 645290365460761, 4236768489465944, 27857102370774193
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 05 2016

Keywords

Links

Crossrefs

Cf. A001764, A026004, A242798.

Programs

  • Maple
    h := n -> hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4):
b := n -> binomial(3*n+1,n)*(n+2)/(2*n+2): # A026004
a := n -> `if`(n=0,1,b(n)*simplify(h(n))):
seq(a(n), n=0..21); # Peter Luschny, Nov 06 2016
  • Mathematica
    f[n_] := (n + 2)Sum[ Binomial[3n - 2i + 1, n - i]/(2n - i + 2), {i, 0, n}]; Array[f, 22, 0] (* Robert G. Wilson v, Nov 06 2016 *)
  • Maxima
    F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(F(x),x)*F(x)/(1-F(x))/x,x,0,10);
  • PARI
    for(n=0,25, print1((n+2)*sum(i=0,n,(binomial(3*n-2*i+1, n-i)/(2*n-i+2))), ", ")) \\ G. C. Greubel, Apr 09 2017

Formula

G.f.: F'(x)*F(x)/(1-F(x))/x, where F(x)/x is g.f. of A001764.
From Vaclav Kotesovec, Nov 06 2016: (Start)
Recurrence: 2*(n+1)*(2*n + 1)*(91*n^4 - 232*n^3 + 15*n^2 + 266*n - 120)*a(n) = (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-1) - (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(91*n^4 + 132*n^3 - 135*n^2 - 36*n + 20)*a(n-3).
a(n) ~ 3^(3*n+7/2) / (7 * sqrt(Pi*n) * 2^(2*n+3)). (End)
a(n) = A026004(n)*hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4). - Peter Luschny, Nov 06 2016

A277957 a(n) = (n+3)*Sum_{i=0..n} binomial(3*n-2*i+2,n-i)/(2*n-i+3).

Original entry on oeis.org

1, 5, 26, 144, 834, 4979, 30361, 188003, 1177694, 7443721, 47384897, 303389530, 1951806313, 12607088771, 81709809546, 531138264252, 3461366814726, 22607751250442, 147952881721126, 969953549401499, 6368831275489633
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 05 2016

Keywords

Links

Crossrefs

Cf. A001764, A242798, A262394, A277956.

Programs

  • Maple
    h := n -> hypergeom([1,-2*n-3,-n],[-3*n/2-1,-3*n/2-1/2],1/4):
b := n -> (n+3)*binomial(3*n+2,n)/(2*n+3): # A262394(n-1)
a := n -> b(n)*simplify(h(n)):
seq(a(n), n=0..21); # Peter Luschny, Nov 06 2016
  • Mathematica
    Table[(n + 3)*Sum[Binomial[3*n - 2*k + 2, n - k]/(2*n - k + 3), {k, 0, n}], {n,0,50}] (* G. C. Greubel, Jun 06 2017 *)
  • Maxima
    F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
taylor(diff(F(x),x)*F(x)^2/(1-F(x))/x^2,x,0,10);
  • PARI
    for(n=0,25, print1((n+3)*sum(k=0,n,binomial(3*n-2*k+2,n-k)/(2*n-k+3)), ", ")) \\ G. C. Greubel, Jun 06 2017

Formula

G.f.: F'(x)*F(x)^2/(1-F(x))/x^2, where F(x)/x is the g.f. of A001764.
From Vaclav Kotesovec, Nov 06 2016: (Start)
Recurrence: 2*(n+1)*(2*n + 3)*(91*n^4 - 24*n^3 - 313*n^2 + 246*n - 24)*a(n) = (2821*n^6 + 5080*n^5 - 12775*n^4 - 5200*n^3 + 8454*n^2 + 420*n - 720)*a(n-1) - (2821*n^6 + 5080*n^5 - 12775*n^4 - 5200*n^3 + 8454*n^2 + 420*n - 720)*a(n-2) + 3*(3*n - 4)*(3*n - 2)*(91*n^4 + 340*n^3 + 161*n^2 - 88*n - 24)*a(n-3).
a(n) ~ 3^(3*n+9/2) / (7 * sqrt(Pi*n) * 2^(2*n+4)). (End)
a(n) = A262394(n-1)*hypergeom([1,-2*n-3,-n],[-3*n/2-1,-3*n/2-1/2],1/4). - Peter Luschny, Nov 06 2016

A323219 a(n) = [x^n] (1 - 4*x)^(-n/2)*x/(1 - x).

Original entry on oeis.org

0, 1, 5, 37, 313, 2811, 26093, 247311, 2377905, 23104441, 226289605, 2230309533, 22093913449, 219786279909, 2194096906461, 21969023675097, 220538907003489, 2218881134793411, 22368588800763701, 225891901214751423, 2284746661102951833, 23140953249273852519
Offset: 0

Views

Author

Peter Luschny, Jan 26 2019

Keywords

Crossrefs

Central diagonal of A323222.
Cf. A242798.

Programs

  • Maple
    ogf := n -> (1 - 4*x)^(-n/2)*x/(1 - x):
ser := n -> series(ogf(n), x, 46):
seq(coeff(ser(n), x, n), n=0..21);

Formula

a(n) = 1/(-3)^(n/2) - 4^n * Pochhammer(n/2,n)/n! * hypergeom([1,3*n/2],[n+1],4). - Robert Israel, Jan 28 2019
From Vaclav Kotesovec, Jan 29 2019: (Start)
Recurrence: 3*(n-2)*(n-1)*(65*n - 213)*a(n) = (20995*n^3 - 152844*n^2 + 347783*n - 238614)*a(n-2) + 12*(3*n - 10)*(3*n - 8)*(65*n - 83)*a(n-4).
a(n) ~ 2^(n - 1/2) * 3^((3*n - 1)/2) / (5*sqrt(Pi*n)). (End)
G.f.: -(24*x*cos(arcsin(216*x^2-1)/3))/(sqrt(3-324*x^2)*(2*sin(arcsin(216*x^2-1)/3)-11)). - Vladimir Kruchinin, Oct 27 2021
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.