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.

A128417 Number triangle T(n,k) = 2^(n-k)*C(2*n,n-k).

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 160, 60, 12, 1, 1120, 448, 112, 16, 1, 8064, 3360, 960, 180, 20, 1, 59136, 25344, 7920, 1760, 264, 24, 1, 439296, 192192, 64064, 16016, 2912, 364, 28, 1, 3294720, 1464320, 512512, 139776, 29120, 4480, 480, 32, 1
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse of A128414. Row sums are A128418. Diagonal sums are A128419.

Examples

			Triangle begins:
  1,
  4, 1,
  24, 8, 1,
  160, 60, 12, 1,
  1120, 448, 112, 16, 1,
  8064, 3360, 960, 180, 20, 1,
  59136, 25344, 7920, 1760, 264, 24, 1,
  439296, 192192, 64064, 16016, 2912, 364, 28, 1
  ...

Links

Crossrefs

Cf. A128413.

Programs

  • Mathematica
    Flatten[Table[2^(n-k) Binomial[2n,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Nov 02 2011 *)

Formula

Riordan array (1/sqrt(1-8*x),(1-4*x-sqrt(1-8*x))/(8*x)).
T(n,k) = 2^(n-k)*A094527(n,k).

A243019 Expansion of -(2*x*sqrt(1-8*x^2)-2*x) / (16*x^3+sqrt(1-8*x^2)*(4*x^2+2*x-1)-8*x^2-2*x+1).

Original entry on oeis.org

1, 1, 5, 7, 33, 51, 233, 379, 1697, 2851, 12585, 21627, 94449, 165075, 714873, 1266027, 5445441, 9746883, 41687369, 75275227, 320420753, 582881971, 2471008281, 4523575371, 19108837601, 35174066851
Offset: 0

Views

Author

Vladimir Kruchinin, May 29 2014

Keywords

Comments

Number of ternary strings of length n that have the same number or more 0's than the combined number of 1's and 2's. For example, a(4) = 33 since the strings are (number of permutations in parentheses): 0000 (1), 0001 (4), 0002 (4), 0011 (6), 0022 (6), 0012 (12). - Enrique Navarrete, Aug 14 2025

Crossrefs

Cf. A000108, A013609, A128418.

Programs

  • Mathematica
    CoefficientList[Series[-(2*x*Sqrt[1-8*x^2]-2*x)/(16*x^3+Sqrt[1-8*x^2]*(4*x^2+2*x-1)-8*x^2-2*x+1),{x,0,20}],x] (* Vaclav Kotesovec, May 29 2014 *)
  • Maxima
    a(n):=sum(2^(i)*binomial(n,i),i,0,floor((n)/2));

Formula

a(n) = sum(i=0..floor(n/2), 2^i*binomial(n,i)).
G.f.: (x*C'(2*x^2))/(C(2*x^2)*(1-x*C(2*x^2))), where C(x) is g.f. of A000108.
a(n) ~ 2^(3*n/2) * (2+sqrt(2) + (-1)^n*(2-sqrt(2))) / sqrt(2*Pi*n). - Vaclav Kotesovec, May 29 2014
D-finite with recurrence: n^2*a(n) = (3*n^2-4)*a(n-1) + 4*(2*n^2 - 2*n - 1)*a(n-2) - 24*(n-2)*(n+1)*a(n-3). - Vaclav Kotesovec, May 29 2014
a(n) = Sum_{k=0..floor(n/2)} A013609(n,k). - Enrique Navarrete, Aug 14 2025

A385252 Number of ternary strings of length 2*n that have at least one 0 but less 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 0, 32, 432, 4608, 45440, 432896, 4051712, 37535744, 345470976, 3165315072, 28905857024, 263303921664, 2393675661312, 21725991600128, 196937443377152, 1783243502256128, 16132632204541952, 145839502212988928, 1317564268289196032, 11896995094093365248, 107375812426273390592
Offset: 0

Views

Author

Enrique Navarrete, Jul 28 2025

Keywords

Examples

			a(2)=32 since the strings of length 4 are (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4).
a(3)=432 since the strings of length 6 are (number of permutations in parentheses): 111110 (6), 111120 (30), 111220 (60), 112220 (60), 122220 (30), 222220 (6), 001111 (15), 001112 (60), 001122 (90), 001222 (60), 002222 (15).

Crossrefs

Cf. A000302, A001019, A128418, A386670.

Programs

  • Mathematica
    a[0]=0; a[n_]:=9^n - 4^n - Sum[2^(n-k)*Binomial[2n,n-k],{k,0,n}]; Array[a,22,0] (* Stefano Spezia, Jul 31 2025 *)

Formula

a(n) = 9^n - 4^n - Sum_{k=0..n} 2^(n-k)*C(2*n,n-k) for n > 0.
G.f.: (5*x*(sqrt(1-8*x))*(sqrt(1-8*x)+12*x-1)-8*x*(36*x^2-13*x+1))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)*(36*x^2-13*x+1)) + 1.
a(n) = A001019(n) - A000302(n) - A128418(n), n > 0.
Conjecture D-finite with recurrence n*a(n) +(-37*n+36)*a(n-1) +4*(131*n-245)*a(n-2) +16*(-221*n+605)*a(n-3) +192*(59*n-213)*a(n-4) +6912*(-2*n+9)*a(n-5)=0. - R. J. Mathar, Jul 31 2025
a(n) = 9^n - 4^n - 2^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/2) for n > 0. - Stefano Spezia, Aug 05 2025

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).

A386826 a(n) = Sum_{k=0..n} 3^(n-k)*C(2*n,n-k).

Original entry on oeis.org

1, 7, 67, 694, 7459, 81922, 912718, 10273228, 116522275, 1329569290, 15244087642, 175472098996, 2026521318286, 23470106563924, 272476942589884, 3169997065488664, 36948020548661539, 431354994430077274, 5043279137171450914, 59041965004582271524, 692026745415822877594, 8119918150063503715324
Offset: 0

Views

Author

Enrique Navarrete, Aug 04 2025

Keywords

Comments

Row sums of number triangle A386825.
Number of strings of length 2*n defined on {0,1,2,3} that have either the same number or more 0's than the combined number of 1's, 2's and 3's.

Examples

			a(3)=694 counts the strings of length 6 as follows: 540 strings with three 0's, 135 with four 0's, 18 with five 0's, and 1 string with six 0's. Hence 694 = 540 + 135 + 18 + 1, where the summands come from the triangle in A386825.

Crossrefs

Cf. A128418, A386825.

Programs

  • Mathematica
    Table[Sum[3^(n-k) Binomial[2n, n-k], {k, 0, n}], {n, 0, 21}]

Formula

a(n) = 3^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/3). - Stefano Spezia, Aug 05 2025
a(n) ~ 2^(2*n-1) * 3^(n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence n*a(n) +(n+1)*a(n-1) +2*(-946*n+2017)*a(n-2) +144*(286*n-821)*a(n-3) +122112*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Aug 26 2025
D-finite with recurrence n*(2*n-1)*a(n) +2*(-28*n^2+20*n+9)*a(n-1) +96*(2*n+1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 26 2025

A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 4, 48, 496, 4864, 46464, 436992, 4068096, 37601280, 345733120, 3166363648, 28910051328, 263320698880, 2393742770176, 21726260035584, 196938517118976, 1783247797223424, 16132649384411136, 145839570932465664, 1317564543167102976, 11896996193604993024, 107375816824319901696
Offset: 0

Views

Author

Enrique Navarrete, Aug 12 2025

Keywords

Examples

			a(2) = 48 since the strings of length 4 are the following (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).

Crossrefs

Cf. A001019, A128418, A385252.

Programs

  • Mathematica
    a[n_] := 9^n - Sum[2^(n-k) * Binomial[2*n, n-k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Aug 16 2025 *)

Formula

a(n) = 9^n - Sum_{k=0..n} 2^(n-k)*binomial(2*n,n-k).
G.f.: (sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)-8*x*(1-9*x))/((1-9*x)*sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A001019(n) - A128418(n).
D-finite with recurrence n*a(n) +(-29*n+28)*a(n-1) +12*(23*n-41)*a(n-2) +432*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025
Showing 1-6 of 6 results.

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