A027476 -id:A027476 - cp's OEIS frontend

A001788 a(n) = n(n+1)2^(n-2).

0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, 860160, 1966080, 4456448, 10027008, 22413312, 49807360, 110100480, 242221056, 530579456, 1157627904, 2516582400, 5452595200, 11777605632, 25367150592, 54492397568, 116769423360, 249644974080, 532575944704

Offset: 0

N. J. A. Sloane

Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube. - Henry Bottomley, Apr 14 2000
Also the number of edges in the (n+1)-halved cube graph. - Eric W. Weisstein, Jun 21 2017
From Philippe Deléham, Apr 28 2004: a(n) is the sum, over all nonempty subsets E of {1, 2, ..., n}, of all elements of E. E.g., a(3) = 24: the nonempty subsets are {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and 1 + 2 + 3 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 2 + 3 = 24.
Equivalently, sum of all nodes (except the last one, equal to n+1) of all integer compositions of n+1. - Olivier Gérard, Oct 22 2011
The inverse binomial transform of a(n-k) for k=-1..4 gives A001844, A000290, A000217(n-1), A002620(n-1), A008805(n-4), A000217 interspersed with 0's. - Michael Somos, Jul 18 2003
Take n points on a finite line. They all move with the same constant speed; they instantaneously change direction when they collide with another; and they fall when they quit the line. a(n-1) is the total number of collisions before falling when the initials directions are the 2^n possible. The mean number of collisions is then n(n-1)/8. E.g., a(1)=0 before any collision is possible. a(2)=1 because there is a collision only if the initials directions are, say, right-left. - Emmanuel Moreau, Feb 11 2006
Also number of pericondensed hexagonal systems with n hexagons. For example, if n=5 then the number of pericondensed hexagonal systems with n hexagons is 24. - Parthasarathy Nambi, Sep 06 2006
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>1, a(n-1) is equal to the number of (n+2)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly two u's. Example: a(2)=6 because we have uuw, uuv, uwu, uvu, wuu and vuu. - Zerinvary Lajos, Dec 29 2007
For n>0 where [0]={}, the empty set, and [n]={1,2,...n} a(n) is the number of ways to separate [n-1] into three non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - Geoffrey Critzer, Feb 07 2009
Form an array with m(n,0) = m(0,n) = n^2 and m(i,j) = m(i-1,j-1) + m(i-1,j). Then m(1,n) = A001844(n) and m(n,n) = a(n). - J. M. Bergot, Nov 07 2012
The sum of the number of inversions of all sequences of zeros and ones with length n+1. - Evan M. Bailey, Dec 09 2020
a(n) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, exactly one 3, and have no restriction on the number of 0s and 1s. For example, a(3)=24 since the strings are 321 (6 of this type), 320 (6 of this type), 310 (6 of this type), 300 (3 of this type) and 311 (3 of this type). - Enrique Navarrete, May 04 2025

			The nodes of an integer composition are the partial sums of its elements, seen as relative distances between nodes of a 1-dimensional polygon. For a composition of 7 such as 1+2+1+3, the nodes are 0,1,3,4,7. Their sum (without the last node) is 8. The sum of all nodes of all 2^(7-1)=64 integer compositions of 7 is 672.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Herbert Izbicki, Über Unterbäume eines Baumes, Monatshefte fur Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Efficient Algorithms for the Bee-Identification Problem, arXiv:2212.09952 [cs.IT], 2022.
Duško Letić, Nenad Cakić, Branko Davidović, Ivana Berković and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, Vol. 2011 (2011), Article 60.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Lara Pudwell, Nathan Chenette and Manda Riehl, Statistics on Hypercube Orientations, AMS Special Session on Experimental and Computer Assisted Mathematics, Joint Mathematics Meetings (Denver 2020).
John Riordan and N. J. A. Sloane, Correspondence, 1974.
R. Tosic, D. Masulovic, I. Stojmenovic, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., Vol. 35, No. 2 (1995), pp. 181-187.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Idempotent Number.
Eric Weisstein's World of Mathematics, Halved Cube Graph.
Eric Weisstein's World of Mathematics, Hypercube Graph.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A001787, A001789, A001793 (sum of all nodes of integer compositions, n included).
Cf. A001844, A038207, A290031 (6-cycles).
Row sums of triangle A094305.
Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), this sequence (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> n*(n+1)*2^(n-2)); # G. C. Greubel, Aug 27 2019

a001788 n = if n < 2 then n else n * (n + 1) * 2 ^ (n - 2)
a001788_list = zipWith (*) a000217_list $ 1 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014

[n*(n+1)*2^(n-2): n in [0..30]]; // G. C. Greubel, Aug 27 2019

A001788 := n->n*(n+1)*2^(n-2);
A001788:=-1/(2*z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero

CoefficientList[Series[x/(1-2x)^3, {x,0,30}], x]
Table[n*(n+1)*2^(n-2), {n,0,30}]
With[{n = 30}, Join[{0}, Times @@@ Thread[{Accumulate[Range[n]], 2^Range[0, n - 1]}]]] (* Harvey P. Dale, Jul 16 2013 *)
LinearRecurrence[{6, -12, 8}, {0, 1, 6}, 30] (* Harvey P. Dale, Jul 16 2013 *)

PARI
```
a(n)=if(n<0,0,2^n*n*(n+1)/4)
```

A001788_upto(n)=Vec(x/(1-2*x)^3+O(x^n),-n) \\ for illustration. - M. F. Hasler, Oct 05 2024

[n if n < 2 else n * (n + 1) * 2**(n - 2) for n in range(28)] # Zerinvary Lajos, Mar 10 2009

G.f.: x/(1-2*x)^3.
E.g.f.: x*(1 + x)*exp(2*x).
a(n) = 2*a(n-1) + n*2^(n-1) = 2*a(n-1) + A001787(n).
a(n) = A038207(n+1,2).
a(n) = A055252(n, 2).
a(n) = Sum_{i=1..n} i^2 * binomial(n, i): binomial transform of A000290. - Yong Kong, Dec 26 2000
a(n) = Sum_{j=0..n} binomial(n+1,j)*(n+1-j)^2. - Zerinvary Lajos, Aug 22 2006
If the leading 0 is deleted, the binomial transform of A001844: (1, 5, 13, 25, 41, ...); = double binomial transform of [1, 4, 4, 0, 0, 0, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = Sum_{1<=i<=k<=n} (-1)^(i+1)*i^2*binomial(n+1,k+i)*binomial(n+1,k-i). - Mircea Merca, Apr 09 2012
a(0)=0, a(1)=1, a(2)=6, a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 16 2013
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(3/2) - 4. (End)

A027472 Third convolution of the powers of 3 (A000244).

Original entry on oeis.org

1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070, 805447196631, 2646469360359, 8661172452084, 28242953648100, 91789599356325, 297398301914493, 960825283108362, 3095992578904722

Offset: 3

Olivier Gérard

Third column of A027465.

With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's. - Zerinvary Lajos, Dec 29 2007

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A000244, A006043, A027465, A075513, A152818.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // G. C. Greubel, May 12 2021

nn=41; Drop[Range[0,nn]!CoefficientList[Series[Exp[x]^3 x^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{9,-27,27}, {1,9,54}, 40] (* G. C. Greubel, May 12 2021 *)
Abs[Take[CoefficientList[Series[1/(1+3x^2)^3,{x,0,60}],x],{1,-1,2}]] (* Harvey P. Dale, Mar 03 2022 *)

a(n)=([0,1,0; 0,0,1; 27,-27,9]^(n-3)*[1;9;54])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[3^(n-3)*binomial(n-1,2) for n in range(3, 40)] # Zerinvary Lajos, Mar 10 2009

Numerators of sequence a[3,n] in (b^2)[i,j]) where b[i,j] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
From Wolfdieter Lang: (Start)
a(n) = 3^(n-3)*binomial(n-1, 2).
G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)
a(n) = |A075513(n, 2)|/9, n >= 3.
a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - Paul Curtz, Jan 07 2009
The sequence 0, 1, 9, 54, ... has e.g.f.: (x + 3*x^2/2)*exp(3*x)/. - Paul Barry, Jul 23 2003
E.g.f.: E(0) where E(k) = 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
With offset=2 e.g.f.: x^2*exp(3*x)/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).
Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)

Corrected by T. D. Noe, Nov 07 2006
Better name from Wolfdieter Lang
Terms a(23) onward added by G. C. Greubel, May 12 2021

A038845 3-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 12, 96, 640, 3840, 21504, 114688, 589824, 2949120, 14417920, 69206016, 327155712, 1526726656, 7046430720, 32212254720, 146028888064, 657129996288, 2937757630464, 13056700579840, 57724360458240, 253987186016256, 1112705767309312, 4855443348258816

Offset: 0

Wolfdieter Lang

Also convolution of A002802 with A000984 (central binomial coefficients).
With a different offset, number of n-permutations of 5 objects u, v, w, z, x with repetition allowed, containing exactly two u's. - Zerinvary Lajos, Dec 29 2007
Also convolution of A000302 with A002697, also convolution of A002457 with itself. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande, Dominik Hangleiter, and Alexey V. Gorshkov, The Second Moment of Hafnians in Gaussian Boson Sampling, arXiv:2403.13878 [quant-ph], 2024. See p. 30.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A000302, A000984, A002457, A002697, A002802, A038231, A052780.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), this sequence (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> 4^n*Binomial(n+2,n) ); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+2, 2): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq((n+2)*(n+1)*4^n/2, n=0..30); # Zerinvary Lajos, Apr 25 2007

Table[4^n*Binomial[n+2,n], {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)

a(n)=(n+2)*(n+1)<<(2*n-1) \\ Charles R Greathouse IV, Aug 21 2015

[4^(n-2)*binomial(n,2) for n in range(2, 30)] # Zerinvary Lajos, Mar 11 2009

a(n) = (n+2)*(n+1)*2^(2*n-1).
G.f.: 1/(1-4*x)^3.
a(n) = Sum_{u+v+w+x+y+z=n} f(u)*f(v)*f(w)*f(x)*f(y)*f(z) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = binomial(n+2,n) * 4^n. - Rui Duarte, Oct 08 2011
E.g.f.: (1 + 8*x + 8*x^2)*exp(4*x). - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 8 - 24*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 40*log(5/4) - 8. (End)

A081139 9th binomial transform of (0,0,1,0,0,0,...).

Original entry on oeis.org

0, 0, 1, 27, 486, 7290, 98415, 1240029, 14880348, 172186884, 1937102445, 21308126895, 230127770466, 2447722649502, 25701087819771, 266895911974545, 2745215094595320, 28001193964872264, 283512088894331673

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001019 (powers of 9).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (27,-243,729).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), this sequence (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

Cf. A001019.

[9^n* Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

LinearRecurrence[{27,-243,729},{0,0,1},30] (* Harvey P. Dale, Jan 30 2018 *)

a(n) = 27*a(n-1) - 243*a(n-2) + 729*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 9^(n-2)*binomial(n, 2).
G.f.: x^2/(1-9*x)^3.
E.g.f.: (x^2/2)*exp(9*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 18 - 144*log(9/8).
Sum_{n>=2} (-1)^n/a(n) = 180*log(10/9) - 18. (End)

A081135 5th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 15, 150, 1250, 9375, 65625, 437500, 2812500, 17578125, 107421875, 644531250, 3808593750, 22216796875, 128173828125, 732421875000, 4150390625000, 23345947265625, 130462646484375, 724792480468750

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000351 (powers of 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), this sequence (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[5^(n-2)*Binomial(n, 2): n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

seq(n*(n-1)*5^(n-2)/2, n=0..30); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[x^2/(1-5x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{15,-75,125},{0,0,1},30] (* Harvey P. Dale, Sep 13 2017 *)

[5^(n-2)*binomial(n,2) for n in range(0, 30)] # Zerinvary Lajos, Mar 12 2009

a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 5^(n-2)*binomial(n, 2).
G.f.: x^2/(1-5*x)^3.
E.g.f.: (x^2/2)*exp(5*x). - G. C. Greubel, May 14 2021
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 40*log(5/4).
Sum_{n>=2} (-1)^n/a(n) = 60*log(6/5) - 10. (End)

A081136 6th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 18, 216, 2160, 19440, 163296, 1306368, 10077696, 75582720, 554273280, 3990767616, 28298170368, 198087192576, 1371372871680, 9403699691520, 63945157902336, 431629815840768, 2894458765049856, 19296391766999040

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, three-fold convolution of A000400 (powers of 6).

Number of n-permutations of 7 objects: p, u, v, w, z, x, y with repetition allowed, containing exactly two u's. - Zerinvary Lajos, May 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (18,-108,216).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), this sequence (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[6^n*Binomial(n+2,2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

seq(binomial(n, 2)*6^(n-2), n=0..19); # Zerinvary Lajos, May 23 2008

nn=20;Range[0,nn]!CoefficientList[Series[x^2/2! Exp[6x],{x,0,nn}],x] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{18,-108,216},{0,0,1},30] (* Harvey P. Dale, Apr 20 2022 *)

[6^(n-2)*binomial(n,2) for n in range(0, 21)] # Zerinvary Lajos, Mar 13 2009

a(n) = 18*a(n-1) -108*a(n-2) +216*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 6^(n-2)*C(n, 2).
G.f.: x^2/(1-6*x)^3.
E.g.f.: exp(6*x) * x^2/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 60*log(6/5).
Sum_{n>=2} (-1)^n/a(n) = 84*log(7/6) - 12. (End)

A027474 a(n) = 7^(n-2) * C(n,2).

Original entry on oeis.org

1, 21, 294, 3430, 36015, 352947, 3294172, 29647548, 259416045, 2219448385, 18643366434, 154231485954, 1259557135291, 10173346092735, 81386768741880, 645668365352248, 5084638377148953, 39779817891812397, 309398583602985310

Offset: 2

Olivier Gérard

7th binomial transform of (0,0,1,0,0,0,........). Starting at 1, the three-fold convolution of A000420 (powers of 7). - Paul Barry, Mar 08 2003

Vincenzo Librandi, Table of n, a(n) for n = 2..400
Index entries for linear recurrences with constant coefficients, signature (21,-147,343).

Third column of A027466.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), this sequence (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[7^(n-2)* Binomial(n, 2): n in [2..20]]; /* Vincenzo Librandi, Oct 12 2011 */

seq(binomial(n, 2)*7^(n-2), n=2..30); # Zerinvary Lajos, Jun 12 2008

Table[7^(n-2) Binomial[n,2], {n,2,20}] (* Harvey P. Dale, Sep 25 2011 *)

a(n)=7^(n-2)*n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

[7^(n-2)*binomial(n,2) for n in range(2, 21)] # Zerinvary Lajos, Mar 13 2009

From Paul Barry, Mar 08 2003: (Start)
G.f.: x^2 / (1-7*x)^3.
a(n) = 21*a(n-1) - 147*a(n-2) + 343*a(n-3), a(0) = a(1) = 0, a(2) = 1. (End)
Numerators of sequence a[3,n] in (a[i,j])^3 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
E.g.f.: (x^2/2)*exp(7*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 14 - 84*log(7/6).
Sum_{n>=2} (-1)^n/a(n) = 112*log(8/7) - 14. (End)

Edited by Ralf Stephan, Dec 30 2004

A081138 8th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 24, 384, 5120, 61440, 688128, 7340032, 75497472, 754974720, 7381975040, 70866960384, 670014898176, 6253472382976, 57724360458240, 527765581332480, 4785074604081152, 43065671436730368, 385057768140177408

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001018 (powers of 8).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-192,512).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), this sequence (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[8^n*Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

LinearRecurrence[{24,-192,512},{0,0,1},30] (* Harvey P. Dale, Jun 08 2014 *)

a(n) = 24*a(n-1) - 192*a(n-2) + 512*a(n-3) for n>2, a(0)=a(1)=0, a(2)=1.
a(n) = 8^(n-2)*binomial(n, 2).
G.f.: x^2/(1 - 8*x)^3.
E.g.f.: (x^2/2)*exp(8*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 16 - 112*log(8/7).
Sum_{n>=2} (-1)^n/a(n) = 144*log(9/8) - 16. (End)

A081140 10th binomial transform of (0,0,1,0,0,0,...).

Original entry on oeis.org

0, 0, 1, 30, 600, 10000, 150000, 2100000, 28000000, 360000000, 4500000000, 55000000000, 660000000000, 7800000000000, 91000000000000, 1050000000000000, 12000000000000000, 136000000000000000, 1530000000000000000

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A011557 (powers of 10).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (30,-300,1000).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), this sequence (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[10^n* Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

Table[10^(n-2)*Binomial[n, 2], {n, 0, 30}] (* G. C. Greubel, May 13 2021 *)

a(n) = 30*a(n-1) - 300*a(n-2) + 1000*a(n-3), a(0)=a(1)=0, a(2)=1.
a(n) = 10^(n-2)*binomial(n, 2).
G.f.: x^2/(1-10*x)^3.
E.g.f.: (x^2/2)*exp(10*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 20 - 180*log(10/9).
Sum_{n>=2} (-1)^n/a(n) = 220*log(11/10) - 20. (End)

A081141 11th binomial transform of (0,0,1,0,0,0,...).

Original entry on oeis.org

0, 0, 1, 33, 726, 13310, 219615, 3382071, 49603708, 701538156, 9646149645, 129687123005, 1711870023666, 22254310307658, 285596982281611, 3624884775112755, 45569980029988920, 568105751040528536

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001020 (powers of 11).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (33,-363,1331).

Cf. A001020.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), this sequence (q=11), A081142 (q=12), A027476 (q=15).

[11^(n-2)*Binomial(n, 2): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

seq((11)^(n-2)*binomial(n,2), n=0..30); # G. C. Greubel, May 13 2021

LinearRecurrence[{33,-363,1331},{0,0,1},30] (* Harvey P. Dale, Dec 15 2014 *)

vector(20, n, n--; 11^(n-2)*binomial(n, 2)) \\ G. C. Greubel, Nov 23 2018

[11^(n-2)*binomial(n, 2) for n in range(20)] # G. C. Greubel, Nov 23 2018

a(n) = 33*a(n-1) - 363*a(n-2) + 1331*a(n-3), a(0) = a(1) = 0, a(2) = 1.
a(n) = 11^(n-2)*binomial(n, 2).
G.f.: x^2/(1 - 11*x)^3.
E.g.f.: (1/2)*exp(11*x)*x^2. - Franck Maminirina Ramaharo, Nov 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 22 - 220*log(11/10).
Sum_{n>=2} (-1)^n/a(n) = 264*log(12/11) - 22. (End)

cp's OEIS Frontend

A001788 a(n) = n*(n+1)*2^(n-2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A027472 Third convolution of the powers of 3 (A000244).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A038845 3-fold convolution of A000302 (powers of 4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A081139 9th binomial transform of (0,0,1,0,0,0,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A081135 5th binomial transform of (0,0,1,0,0,0, ...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A081136 6th binomial transform of (0,0,1,0,0,0, ...).

Views

Author

Keywords

Comments

Links

Crossrefs

A001788 a(n) = n(n+1)2^(n-2).