A129953 -id:A129953 - cp's OEIS frontend

A129952 Binomial transform of A124625.

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

Paul Curtz, Jun 10 2007

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

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

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

LinearRecurrence[{4, -4}, {1, 1, 2, 6}, 30] (* G. C. Greubel, Jun 08 2016; corrected by Georg Fischer, Apr 02 2019 *)

{m=29; print1(1, ",", 1, ","); for(n=2, m, print1(n*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

def A129952(n): return n<1 else 1 # Chai Wah Wu, Oct 03 2024

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

Edited and extended by Klaus Brockhaus, Jun 17 2007

A129954 Second differences of A129952.

Original entry on oeis.org

1, 3, 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

Paul Curtz, Jun 10 2007

First differences of A129953: a(n) = A129953(n+1) - A129953(n).

Essentially the same as A078836: a(n) = A078836(n+4) for n > 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A129952, A129953, A078836.

m:=16; 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] ]; [ U[n+1]-U[n]: n in[1..2*m-2] ]; // Klaus Brockhaus, Jun 17 2007

{m=29; print1(1, ",", 3, ","); for(n=2, m, print1((n+4)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

def A129954(n): return n+4<1 else 2*n+1 # Chai Wah Wu, Oct 03 2024

a(0) = 1, a(1) = 3; for n > 1, a(n) = (n+4)*2^(n-2).
G.f.: (1-x)*(1-2*x^2)/(1-2*x)^2.
Binomial transform of [1, 2, 1, 4, 1, 6, 1, 8, ...]. - Gary W. Adamson, Sep 29 2007
E.g.f.: (x + exp(2*x)*(2 + x))/2. - Stefano Spezia, Oct 04 2024

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

Paul Curtz, Jun 10 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -4).

Cf. A129952, A129953, A129954, A034007.

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

Differences[LinearRecurrence[{4,-4},{1,1,2,6},40],3] (* Harvey P. Dale, Sep 04 2020 *)

{m=29; print1(2, ",", 3, ","); for(n=2, m, print1((n+6)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

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)

Edited and extended by Klaus Brockhaus, Jun 17 2007

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

Paul Curtz, Jun 15 2007

nonn

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.

			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.

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)

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

A360951 Expansion of e.g.f. (cosh(x) - 1)(1 + x)exp(x).

Original entry on oeis.org

0, 0, 1, 6, 19, 50, 121, 280, 631, 1398, 3061, 6644, 14323, 30706, 65521, 139248, 294895, 622574, 1310701, 2752492, 5767147, 12058602, 25165801, 52428776, 109051879, 226492390, 469762021, 973078500, 2013265891, 4160749538, 8589934561, 17716740064, 36507221983, 75161927646, 154618822621

Offset: 0

Enrique Navarrete, Feb 26 2023

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first set has an even number of elements greater than or equal to two, the second set has either one or no elements, and the third set has no restrictions.

			The 19 set partitions for n=4 are the following:
  {1,2,3,4}, { }, { } (one of these);
  {1,2}, { }, {3,4}   (6 of these);
  {1,2}, {3}, {4}     (12 of these).

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A129953.

a(n) = 2^(n-1) + n*2^(n-2) - n - 1 for n >= 2.
G.f.: x^2*(1 - 4*x^2 + 2*x^3)/((1 - x)^2*(1 - 2*x)^2). - Stefano Spezia, Mar 04 2023
a(n) = (n + 2)*2^(n-2) - n - 1 = A129953(n) - n - 1 for n >= 2. - Stefano Spezia, Mar 05 2023

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A129952 Binomial transform of A124625.

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A129954 Second differences of A129952.

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A129955 Third differences of A129952.

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

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A360951 Expansion of e.g.f. (cosh(x) - 1)*(1 + x)*exp(x).

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A360951 Expansion of e.g.f. (cosh(x) - 1)(1 + x)exp(x).