A152194 -id:A152194 - cp's OEIS frontend

A152195 Triangle read by rows, A000012 * A152194.

1, 2, 4, 1, 8, 4, 16, 11, 1, 32, 26, 6, 64, 57, 22, 1, 128, 120, 64, 8, 256, 247, 163, 37, 1, 512, 502, 382, 130, 10, 1024, 1013, 848, 386, 56, 1, 2048, 2036, 1816, 1024, 232, 12, 4096, 4083, 3797, 2510, 794, 79, 1

Offset: 0

Gary W. Adamson, Nov 28 2008

Row sums = A045623: (1, 2, 5, 12, 28, 64, 144,...).

			First few rows of the triangle =
1;
2;
4, 1;
8, 4;
16, 11, 1;
32, 26, 6;
64, 57, 22, 1;
128, 120, 64, 8,
256, 247, 163, 37, 1;
512, 502, 382, 130, 10;
1024, 1013, 848, 386, 56, 1;
2048, 2036, 1816, 1024, 232, 12;
4096, 4083, 3797, 2510, 794, 79, 1;
...

A152194, A045623

Triangle read by rows, A000012 * A152194; A000012 = an infinite lower triangular matrix with all 1's. The operation performs partial sums of A152194 column terms.

Equals binomial transform of [1; 1; 1,1; 1,1; 1,1,1; 1,1,1;...], (an infinite lower triangular matrix with one 1 in two rows, two 1's in two rows, three 1's in two rows, etc...).

A045891 First differences of A045623.

Original entry on oeis.org

1, 1, 3, 7, 16, 36, 80, 176, 384, 832, 1792, 3840, 8192, 17408, 36864, 77824, 163840, 344064, 720896, 1507328, 3145728, 6553600, 13631488, 28311552, 58720256, 121634816, 251658240, 520093696, 1073741824, 2214592512, 4563402752

Offset: 0

Wolfdieter Lang

Let M_n be the n X n matrix m_(i,j) = 3 + abs(i-j), then det(M_n) =(-1)^(n+1)*a(n+1). - Benoit Cloitre, May 28 2002
If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1..n). - Milan Janjic, Nov 18 2007
Equals row sums of triangle A152194. - Gary W. Adamson, Nov 28 2008
An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A045623. - Johannes W. Meijer, Aug 15 2010
a(n) is the total number of runs of 1 in the compositions of n+1. For example, a(3) = A045623(3) - A045623(2) = 12 - 5 = 7 runs of only 1 in the compositions of 4, enumerated "()" as follows: 3,(1); (1),3; 2,(1,1);(1),2,(1); (1,1),2; (1,1,1,1). More generally, the total number of runs of only part k in the compositions of n+k is A045623(n) - A045623(n-k). - Gregory L. Simay, May 02 2017
This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^2 + S^3; see A291000.  - Clark Kimberling, Aug 24 2017

			G.f. = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 36*x^5 + 80*x^6 + ... - _Michael Somos_, Mar 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Frank Ellermann, Illustration of binomial transforms
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Thomas Selig and Haoyue Zhu, New combinatorial perspectives on MVP parking functions and their outcome map, arXiv:2309.11788 [math.CO], 2023. See p. 29.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A027656, A034007, A045623, A152194, A175655.

[1,1] cat [(n+4)*2^(n-3): n in [2..40]]; // G. C. Greubel, Sep 27 2022

Join[{1,1,a=3,b=7},Table[c=4*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
Table[ If[n<2, 1, 2^(n-3)*(n+4)], {n, 0, 30}] (* Jean-François Alcover, Sep 12 2012 *)
LinearRecurrence[{4,-4},{1,1,3,7},40] (* Harvey P. Dale, May 03 2019 *)

v=[1,1,3,7];for(i=1,99,v=concat(v,4*(v[#v]-v[#v-1])));v \\ Charles R Greathouse IV, Jun 01 2011

[1,1]+[(n+4)*2^(n-3) for n in range(2,40)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k=0..n-2} (k+3)*binomial(n-2,k) for n >= 2. - N. J. A. Sloane, Jan 30 2008
a(n) = (n+4)*2^(n-3), n >= 2, with a(0) = a(1) = 1.
G.f.: (1-x)^3/(1-2*x)^2.
Equals binomial transform of A027656.
Starting 1, 3, 7, 16, ... this is ((n+5)*2^n - 0^n)/4, the binomial transform of (1, 2, 2, 3, 3, ...). - Paul Barry, May 20 2003
From Paul Barry, Nov 29 2004: (Start)
a(n) = ((n+4)*2^(n-1) + 3*C(0, n) - C(1, n))/4;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*(k+1). (End)
a(n) = A045623(n-1) + 2^(n-2) = A034007(n+1) - 2^(n-2) for n>=2. - Philippe Deléham, Apr 20 2009
G.f.: 1 + Q(0)*x/(1-x)^2, where Q(k)= 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013
a(n) = Sum_{k=0..n} (k+1)*C(n-2,n-k). Peter Luschny, Apr 20 2015
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=0} 1/a(n) = 128*log(2) - 1292/15.
Sum_{n>=0} (-1)^n/a(n) = 782/15 - 128*log(3/2). (End)
E.g.f.: (2 - x + exp(2*x)*(2 + x))/4. - Stefano Spezia, Mar 26 2022

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A152195 Triangle read by rows, A000012 * A152194.

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A045891 First differences of A045623.

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