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 1-6 of 6 results.

A128418 a(n) = Sum_{k=0..n} 2^(n-k)*C(2n,n-k).

Original entry on oeis.org

1, 5, 33, 233, 1697, 12585, 94449, 714873, 5445441, 41687369, 320420753, 2471008281, 19108837601, 148123058153, 1150532419377, 8952614975673, 69772391628417, 544532315255433, 4255064364533457, 33287174505889113, 260669265451935777, 2043172307192457513, 16028314647309873777
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Row sums of number triangle A128417.

Links

Programs

  • Mathematica
    Table[Sum[2^(n-k) Binomial[2n,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 06 2013 *)
  • PARI
    x='x +O('x^50); Vec(8*x/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1))) \\ G. C. Greubel, Feb 09 2017

Formula

G.f.: 8*x/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1));
D-finite with recurrence n^2*a(n)+(12+4*n-17*n^2)*a(n-1) +36*(n+1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 05 2012
a(n) ~ 2^(3*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 03 2014

A128414 Riordan array ((1-2x)/(1+2x),x/(1+2x)^2).

Original entry on oeis.org

1, -4, 1, 8, -8, 1, -16, 36, -12, 1, 32, -128, 80, -16, 1, -64, 400, -400, 140, -20, 1, 128, -1152, 1680, -896, 216, -24, 1, -256, 3136, -6272, 4704, -1680, 308, -28, 1, 512, -8192, 21504, -21504, 10560, -2816
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse is A128417. Row sums are A128415. Diagonal sums are A128416.

Examples

			Triangle begins
1;
-4, 1;
8, -8, 1;
-16, 36, -12, 1;
32, -128, 80, -16, 1;
-64, 400, -400, 140, -20, 1;
128, -1152, 1680, -896, 216, -24, 1;
-256, 3136, -6272, 4704, -1680, 308, -28, 1;
512, -8192, 21504, -21504, 10560, -2816, 416, -32, 1;

Crossrefs

Cf. A128413.

Formula

T(n,k) = 2^(n-k)*A110162(n,k). - Philippe Deléham, Jan 18 2014
T(n,k) = T(n-1,k-1) - 4*T(n-1,k) - 4*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = -4, T(2,0) = 8, T(2,1) = -8, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 18 2014

A128413 Inverse of number triangle A128412.

Original entry on oeis.org

1, 2, 1, 12, 8, 1, 80, 60, 12, 1, 560, 448, 112, 16, 1, 4032, 3360, 960, 180, 20, 1, 29568, 25344, 7920, 1760, 264, 24, 1, 219648, 192192, 64064, 16016, 2912, 364, 28, 1, 1647360, 1464320, 512512, 139776, 29120, 4480, 480, 32, 1, 12446720
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

First column is A069723.

Examples

			Triangle begins
1,
2, 1,
12, 8, 1,
80, 60, 12, 1,
560, 448, 112, 16, 1,
4032, 3360, 960, 180, 20, 1,
29568, 25344, 7920, 1760, 264, 24, 1,
219648, 192192, 64064, 16016, 2912, 364, 28, 1

Crossrefs

Cf. A128417.

Formula

Number triangle T(n,k)=if(k=0,2^n*(C(2n,n)/2+0^n/2),2^(n-k)*C(2n,n-k)); Column k has g.f. if(k=0,4x/(sqrt(1-8x)(1-sqrt(1-8x))),(1/sqrt(1-8x))*((1-4x-sqrt(1-8x))/(8x))^k);

A128419 Expansion of 8/(sqrt(1-8*x)*(sqrt(1-8*x)+4*x+7)).

Original entry on oeis.org

1, 4, 25, 168, 1181, 8524, 62609, 465616, 3495013, 26423604, 200920985, 1534936440, 11771854381, 90578698396, 698921030945, 5406132020128, 41905249405301, 325434733291396, 2531523208218665, 19721766268370248, 153847524455503421, 1201601094053039596, 9395224234956935345
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Diagonal sums of number triangle A128417.

Links

Programs

  • Mathematica
    CoefficientList[Series[8/(Sqrt[1-8x](Sqrt[1-8x]+4x+7)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2012 *)
Table[Sum[2^(n-2*k)*Binomial[2*n-2*k,n-2*k], {k,0,Floor[n/2]}],{n,0,50}] (* G. C. Greubel, Feb 09 2017 *)
  • PARI
    x='x+O('x^50); Vec(8/(sqrt(1-8*x)*(sqrt(1-8*x)+4*x+7))) \\ G. C. Greubel, Feb 09 2017

Formula

a(n) = Sum_{k=0..floor(n/2)} 2^(n-2k)*C(2n-2k,n-2k).
D-finite with recurrence: 3*n*(15*n-22)*a(n) = 4*(75*n^2-155*n+63)*a(n-1) + (465*n^2-922*n+336)*a(n-2) + 4*(2*n-3)*(15*n-7)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(3*n+4)/(15*sqrt(Pi*n)) . - Vaclav Kotesovec, Oct 20 2012

A386670 Number of ternary strings of length 2*n that have more 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 1, 9, 73, 577, 4521, 35313, 275577, 2150721, 16793929, 131230609, 1026283545, 8032614625, 62921342953, 493262044977, 3869724080313, 30379987189377, 238661880787593, 1876072096450257, 14756076838714713, 116126703647975457, 914363729294862633, 7203083947383222897
Offset: 0

Views

Author

Enrique Navarrete, Jul 28 2025

Keywords

Examples

			a(1)=1 since the string of length 2 is 00.
a(2)=9 since the strings of length 4 are the 4 permutations of 0001, the 4 permutations of 0002, and 0000.
a(4)=577 since the strings of length 8 are (number of permutations in parentheses): 00000001 (8), 00000002 (8), 00000011 (28), 00000012 (56), 00000022 (28), 00000111 (56), 00000112 (168), 00000122 (168), 00000222 (56), 00000000 (1).

Crossrefs

Cf. A054554, A059304, A128417, A128418, A243019.

Formula

a(n) = Sum_{k=1..n} 2^(n-k)*binomial(2*n,n-k).
a(n) = Sum_{k=1..n} A128417(n,k).
G.f.: (1-4*x-sqrt(1-8*x))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A128418(n) - A059304(n).

A386825 Triangle read by rows: T(n,k) = 3^(n-k)*C(2*n,n-k).

Original entry on oeis.org

1, 6, 1, 54, 12, 1, 540, 135, 18, 1, 5670, 1512, 252, 24, 1, 61236, 17010, 3240, 405, 30, 1, 673596, 192456, 40095, 5940, 594, 36, 1, 7505784, 2189187, 486486, 81081, 9828, 819, 42, 1, 84440070, 25019280, 5837832, 1061424, 147420, 15120, 1080, 48, 1, 956987460, 287096238
Offset: 0

Views

Author

Enrique Navarrete, Aug 04 2025

Keywords

Comments

Row sums are A386826.

Examples

			Triangle begins:
        1;
        6,       1;
       54,      12,      1;
      540,     135,     18,     1;
     5670,    1512,    252,    24,    1;
    61236,   17010,   3240,   405,   30,   1;
   673596,  192456,  40095,  5940,  594,  36,  1;
  7505784, 2189187, 486486, 81081, 9828, 819, 42, 1;
  ...

Crossrefs

Cf. A094527, A128417, A386826.

Programs

  • Mathematica
    Flatten[Table[3^(n-k) Binomial[2n, n-k], {n, 0, 9}, {k, 0, n}]]

Formula

T(n,k) = 3^(n-k)*A094527(n,k).
Showing 1-6 of 6 results.

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