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 1-5 of 5 results.

A129952 Binomial transform of A124625.

Original entry on oeis.org

1, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112
Offset: 0

Views

Author

Paul Curtz, Jun 10 2007

Keywords

Comments

Essentially the same as A057711: a(n) = A057711(n) for n >= 1.
Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

Links

Crossrefs

Cf. A124625, A045623, A057711, A129953 (first differences), A129954 (second differences), A129955 (third differences).

Programs

  • Magma
    m:=15; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; [ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; // Klaus Brockhaus, Jun 17 2007
  • Mathematica
    LinearRecurrence[{4, -4}, {1, 1, 2, 6}, 30] (* G. C. Greubel, Jun 08 2016; corrected by Georg Fischer, Apr 02 2019 *)
  • PARI
    {m=29; print1(1, ",", 1, ","); for(n=2, m, print1(n*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007
  • Python
    def A129952(n): return n<1 else 1 # Chai Wah Wu, Oct 03 2024

Formula

a(0) = 1, a(1) = 1; for n > 1, a(n) = n*2^(n-2).
G.f.: (1-3*x+2*x^2+2*x^3)/(1-2*x)^2.
E.g.f.: (1/2)*(x*exp(2*x) + x + 2). - G. C. Greubel, Jun 08 2016

Extensions

Edited and extended by Klaus Brockhaus, Jun 17 2007

A129955 Third differences of A129952.

Original entry on oeis.org

2, 3, 8, 18, 40, 88, 192, 416, 896, 1920, 4096, 8704, 18432, 38912, 81920, 172032, 360448, 753664, 1572864, 3276800, 6815744, 14155776, 29360128, 60817408, 125829120, 260046848, 536870912, 1107296256, 2281701376, 4697620480
Offset: 0

Views

Author

Paul Curtz, Jun 10 2007

Keywords

Links

Crossrefs

Cf. A129952, A129953, A129954, A034007.

Programs

  • Magma
    m:=17; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; T:=[ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; U:=[ T[n+1]-T[n]: n in[1..2*m-1] ]; V:=[ U[n+1]-U[n]: n in[1..2*m-2] ]; [ V[n+1]-V[n]: n in[1..2*m-3] ]; // Klaus Brockhaus, Jun 17 2007
  • Mathematica
    Differences[LinearRecurrence[{4,-4},{1,1,2,6},40],3] (* Harvey P. Dale, Sep 04 2020 *)
  • PARI
    {m=29; print1(2, ",", 3, ","); for(n=2, m, print1((n+6)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

Formula

First differences of A129954: a(n) = A129954(n+1) - A129954(n).
a(n) = A034007(n+2)-2^(n-2) for n > 1.
a(0) = 2, a(1) = 3; for n > 1, a(n) = (n+6)*2^(n-2).
G.f.: (2-5*x+4*x^2-2*x^3)/(1-2*x)^2.
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=0} 1/a(n) = 256*log(2) - 12347/70.
Sum_{n>=0} (-1)^n/a(n) = 21851/210 - 256*log(3/2). (End)

Extensions

Edited and extended by Klaus Brockhaus, Jun 17 2007

A131352 Row sums of triangle A133935.

Original entry on oeis.org

1, 2, 6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024
Offset: 0

Views

Author

Gary W. Adamson, Sep 29 2007

Keywords

Examples

			a(3) = 14 = sum of row 3 terms of triangle A133935: (1 + 3 + 9 + 1); = (1, 3, 3, 1) dot (1, 1, 3, 1).

Links

Crossrefs

Cf. A133935, A093178.

Programs

  • Mathematica
    CoefficientList[Series[(1-2x+2x^2-2x^3)/(1-2x)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-4},{1,2,6,14},40] (* Harvey P. Dale, Dec 04 2021 *)

Formula

Binomial transform of A093178: (1, 1, 3, 1, 5, 1, 7, 1...)
a(n) = A129954(n), n>1. G.f.: (1-2x+2x^2-2x^3)/(1-2x)^2. [R. J. Mathar, Dec 13 2008]
a(n) = 2^(n-2)*(n+4) for n>1. - _Colin Barker, Jun 05 2012

Extensions

Extended by R. J. Mathar, Dec 13 2008

A130002 Alternating sum along antidiagonals of the array of A129952 and its iterated differences.

Original entry on oeis.org

1, 1, 2, 3, 10, 23, 60, 139, 326, 735, 1648, 3635, 7962, 17287, 37316, 80091, 171118, 364079, 771864, 1631107, 3436994, 7223511, 15146092, 31690283, 66176790, 137945983, 287076800, 596523219, 1237785706, 2565049895, 5309056788, 10976027515, 22667882942
Offset: 0

Views

Author

Paul Curtz, Jun 15 2007

Keywords

Comments

Define the square array T of A129952 and its iterated differences: T(0,n)=A129952(n), T(d,n)=T(d-1,n+1)-T(d-1,n), d>0. Then a(n) = sum_{d=0..n} (-1)^d*T(d,n-d), the sum along the antidiagonals of T(d,n), alternating signs.

Examples

			The original series and first, 2nd etc. differences are the rows of
1..1..2...6..16..40.. <- A129952 = T(0,n)
0..1..4..10..24..56.. <- A129953 = T(1,n)
1..3..6..14..32..72.. <- A129954 = T(2,n)
2..3..8..18..40..88.. <- A129955 = T(3,n)
1..5.10..22..48......
...
a(2) = 2-1+1 = 2. a(3) = 6-4+3-2 = 3. a(4) = 16-10+6-3+1 = 10.

Formula

From Chai Wah Wu, Jan 30 2018: (Start)
a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 7*a(n-4) + 4*a(n-5) - 4*a(n-6) for n > 6.
G.f.: (-2*x^6 - 6*x^5 + 3*x^4 + 5*x^3 - 3*x + 1)/((x - 1)^2*(x + 1)^2*(2*x - 1)^2). (End)

Extensions

Edited and extended by R. J. Mathar, Jun 30 2008

A133934 A007318 * (a diagonalized version of A124625).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 1, 6, 3, 4, 1, 8, 6, 16, 1, 1, 10, 10, 40, 5, 6, 1, 12, 15, 80, 15, 36, 1, 1, 14, 21, 140, 35, 126, 7, 8, 1, 16, 28, 224, 70, 336, 28, 64, 1, 1, 18, 36, 336, 126, 756, 84, 288, 9, 10
Offset: 1

Views

Author

Gary W. Adamson, Sep 29 2007

Keywords

Comments

Row sums = A129954: (1, 3, 6, 14, 32, 72, 160, ...).

Examples

			First few rows of the triangle:
  1;
  1,  2;
  1,  4,  1;
  1,  6,  3,  4;
  1,  8,  6, 16,  1;
  1, 10, 10, 40,  5,  6;
  1, 12, 15, 80, 15, 36,  1;
  ...

Crossrefs

Cf. A129954.

Formula

Binomial transform of an infinite lower triangular matrix, with (1, 2, 1, 4, 1, 6, ...) in the main diagonal and zeros elsewhere.
Showing 1-5 of 5 results.

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