A047926 -id:A047926 - cp's OEIS frontend

A094176 (A047926(n)-A091588(n))/2.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 15, 43, 127, 383, 1143, 3423, 10264, 30785

Offset: 0

N. J. A. Sloane, Jun 01 2004

nonn

A091588 remains mysterious, but A047926 is reasonably close to it.

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
Index entries for sequences related to Gijswijt's sequence

Cf. A090822, A047926, A091588.

A005917 Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

Original entry on oeis.org

1, 15, 65, 175, 369, 671, 1105, 1695, 2465, 3439, 4641, 6095, 7825, 9855, 12209, 14911, 17985, 21455, 25345, 29679, 34481, 39775, 45585, 51935, 58849, 66351, 74465, 83215, 92625, 102719, 113521, 125055, 137345, 150415, 164289, 178991

Offset: 1

N. J. A. Sloane

Final digits of a(n), i.e., a(n) mod 10, are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), i.e., a(n) mod 100, are repeated periodically with period of length 50. - Alexander Adamchuk, Aug 11 2006
a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny, Feb 26 2007
If Y is a 3-subset of a 2n-set X then, for n >= 2, a(n-2) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment in A006003. A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. - David Quentin Dauthier, Nov 07 2008
Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). - J. M. Bergot, Jun 25 2013
Bisection of A006003. - Omar E. Pol, Sep 01 2018
Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49, ...]; [3, 5, 15, 29, 47, ...]; [9, 11, 13, 27, 45, ...]; [19, 21, 23, 25, 43, ...]; [33, 35, 37, 39, 41, ...]. - J. M. Bergot, Jul 16 2013 [This contribution was moved here from A047926 by Petros Hadjicostas, Mar 08 2021.]
For n>=2, these are the primitive sides s of squares of type 2 described in A344332. - Bernard Schott, Jun 04 2021
(a(n) + 1) / 2 = A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - George Sicherman, Jan 21 2024

J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.
E. Deza and M. M. Deza, Figurate Numbers, World Scientific Publishing, 2012, pp. 123-124.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Mario Defranco and Paul E. Gunnells, Hypergraph matrix models and generating functions, arXiv:2204.11361 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (9).
Andy Nicol, Illustration of Rhombic Dodecahedral Numbers
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7
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
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Eric Weisstein's World of Mathematics, Nexus Number.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A063493, A063494, A063495, A063496.
Column k=3 of A047969.
Cf. A128195, A176850, A005408, A176271, A212133.
Cf. A001844, A000583, A000290.
Cf. A007588, A000292, A000332, A002817, A342354.
Cf. A031215, A008292.
Cf. A016754, A344330, A344332.

a005917 n = a005917_list !! (n-1)
a005917_list = map sum $ f 1 [1, 3 ..] where
   f x ws = us : f (x + 2) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, Nov 13 2014

[n^4 - (n-1)^4: n in [1..50]]; // Vincenzo Librandi, Aug 01 2011

Table[n^4-(n-1)^4,{n,40}]  (* Harvey P. Dale, Apr 01 2011 *)
#[[2]]-#[[1]]&/@Partition[Range[0,40]^4,2,1] (* More efficient than the above Mathematica program because it only has to calculate each 4th power once *) (* Harvey P. Dale, Feb 07 2015 *)
Differences[Range[0,40]^4] (* Harvey P. Dale, Aug 11 2023 *)

a(n)=n^4-(n-1)^4 \\ Charles R Greathouse IV, Jul 31 2011

A005917_list, m = [], [24, -12, 2, 1]
for _ in range(10**2):
    A005917_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

a(n) = (2*n - 1)*(2*n^2 - 2*n + 1).
Sum_{i=1..n} a(i) = n^4 = A000583(n). First differences of A000583.
G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
More generally, g.f. for n^m - (n - 1)^m is Euler(m, x)/(1 - x)^m, where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292). E.g.f.: x*(exp(y/(1 - x)) - exp(x*y/(1 - x)))/(exp(x*y/(1 - x))-x*exp(y/(1 - x))). - Vladeta Jovovic, May 08 2002
a(n) = sum of the next (2*n - 1) odd numbers; i.e., group the odd numbers so that the n-th group contains (2*n - 1) elements like this: (1), (3, 5, 7), (9, 11, 13, 15, 17), (19, 21, 23, 25, 27, 29, 31), ... E.g., a(3) = 65 because 9 + 11 + 13 + 15 + 17 = 65. - Xavier Acloque, Oct 11 2003
a(n) = 2*n - 1 + 12*Sum_{i = 1..n} (i - 1)^2. - Xavier Acloque, Oct 16 2003
a(n) = (4*binomial(n,2) + 1)*sqrt(8*binomial(n,2) + 1). - Paul Barry, Mar 14 2004
Binomial transform of [1, 14, 36, 24, 0, 0, 0, ...], if the offset is 0. - Gary W. Adamson, Dec 20 2007
Sum_{i=1..n-1}(a(i) + a(i+1)) = 8*Sum_{i=1..n}(i^3 + i) = 16*A002817(n-1) for n > 1. - Bruno Berselli, Mar 04 2011
a(n+1) = a(n) + 2*(6*n^2 + 1) = a(n) + A005914(n). - Vincenzo Librandi, Mar 16 2011
a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n) - A181475(n-2)). - Bruno Berselli, Sep 26 2011
a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n - 1). - Reinhard Zumkeller, Jan 18 2012
a(n+1) = Sum_{k=0..2*n+1} (A176850(n,k) - A176850(n-1,k))*(2*k + 1), n >= 1. - L. Edson Jeffery, Nov 02 2012
a(n) = A005408(n-1) * A001844(n-1) = (2*(n - 1) + 1) * (2*(n - 1)*n + 1) = A000290(n-1)*12 + 2 + a(n-1). - Bruce J. Nicholson, May 17 2017
a(n) = A007588(n) + A007588(n-1) = A000292(2n-1) + A000292(2n-2) + A000292(2n-3) = A002817(2n-1) - A002817(2n-2). - Bruce J. Nicholson, Oct 22 2017
a(n) = A005898(n-1) + 6*A000330(n-1) (cf. Deza, Deza, 2012, p. 123, Section 2.6.2). - Felix Fröhlich, Oct 01 2018
a(n) = A300758(n-1) + A005408(n-1). - Bruce J. Nicholson, Apr 23 2020
G.f.: polylog(-4, x)*(1-x)/x. See the Simon Plouffe formula above (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

A094195 Expansion of g.f.: (1-4x)/((1-5x)*(1-x)^2).

Original entry on oeis.org

1, 3, 10, 42, 199, 981, 4888, 24420, 122077, 610359, 3051766, 15258798, 76293955, 381469737, 1907348644, 9536743176, 47683715833, 238418579115, 1192092895522, 5960464477554, 29802322387711, 149011611938493, 745058059692400, 3725290298461932, 18626451492309589

Offset: 0

N. J. A. Sloane, Jun 01 2004

An approximation to A091843.

G. C. Greubel, Table of n, a(n) for n = 0..1000
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
Index entries for sequences related to Gijswijt's sequence
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A047926, A073724, A090822, A091843.

A row of A094250.

[(5^(n+1) +12*n +11)/16: n in [0..40]]; // G. C. Greubel, Aug 18 2023

CoefficientList[Series[(1-4x)/((1-5x)(1-x)^2),{x,0,30}],x] (* or *) LinearRecurrence[{7,-11,5},{1,3,10},30] (* Harvey P. Dale, Dec 31 2011 *)

[(5^(n+1) +12*n +11)/16 for n in range(41)] # G. C. Greubel, Aug 18 2023

a(n) = (5^(n+1) + 12*n + 11)/16.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3), with a(0)=1, a(1)=3, a(2)=10. - Harvey P. Dale, Dec 31 2011
E.g.f.: (1/16)*(5*exp(5*x) + (11 + 12*x)*exp(x)). - G. C. Greubel, Aug 18 2023

A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)n(n+1) - 1)/(n+1)^2, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 8, 15, 1, 3, 9, 22, 31, 1, 3, 10, 31, 63, 63, 1, 3, 11, 42, 117, 185, 127, 1, 3, 12, 55, 199, 459, 550, 255, 1, 3, 13, 70, 315, 981, 1825, 1644, 511, 1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023, 1, 3, 15, 106, 673, 3273, 11203, 24420, 29133, 14767, 2047

Offset: 0

N. J. A. Sloane, Jun 02 2004

			Array, A(n, k), begins:
  1, 3,  7, 15,  31,   63,   127,    255,     511, ... A000225;
  1, 3,  8, 22,  63,  185,   550,   1644,    4925, ... A047926;
  1, 3,  9, 31, 117,  459,  1825,   7287,   29133, ... A073724;
  1, 3, 10, 42, 199,  981,  4888,  24420,  122077, ... A094195;
  1, 3, 11, 55, 315, 1871, 11203,  67191,  403115, ... A094259;
  1, 3, 12, 70, 471, 3273, 22882, 160140, 1120941, ...
Antidiagonals, T(n, k), begins as:
  1;
  1, 3;
  1, 3,  7;
  1, 3,  8, 15;
  1, 3,  9, 22,  31;
  1, 3, 10, 31,  63,   63;
  1, 3, 11, 42, 117,  185,  127;
  1, 3, 12, 55, 199,  459,  550,  255;
  1, 3, 13, 70, 315,  981, 1825, 1644,  511;
  1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows are A000225, A047926, A073724, A094195, A094259.

A094250:= func< n,k | ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 >;
[A094250(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A094250[n_, k_]:= ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2;
Table[A094250[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A094250(n, k): return ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2
flatten([[A094250(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2 (array).
T(n, k) = ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 (antidiagonals).
G.f. for row n: (1-(n+1)*x)/((1-(n+2)*x)*(1-x)^2).

A342354 M(n,k) = 2n^2 + 2k + 1 for 0 <= k <= n and M(n,k) = 2k^2 + 4k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

Original entry on oeis.org

1, 3, 7, 9, 5, 17, 19, 11, 15, 31, 33, 21, 13, 29, 49, 51, 35, 23, 27, 47, 71, 73, 53, 37, 25, 45, 69, 97, 99, 75, 55, 39, 43, 67, 95, 127, 129, 101, 77, 57, 41, 65, 93, 125, 161, 163, 131, 103, 79, 59, 63, 91, 123, 159, 199, 201, 165, 133, 105, 81, 61, 89, 121, 157, 197, 241, 243, 203, 167, 135, 107, 83, 87, 119, 155, 195, 239, 287

Offset: 0

Petros Hadjicostas, Mar 08 2021

This is a square array defined by J. M. Bergot in A005917 (originally by mistake in A047926). Here is the edited description of the array by this contributor.

Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals A005917(n+1).

			Square array M(n,k) (n, k >= 0) begins:
   1,  7, 17, 31, 49, 71, 97, 127, ...
   3,  5, 15, 29, 47, 69, 95, 125, ...
   9, 11, 13, 27, 45, 67, 93, 123, ...
  19, 21, 23, 25, 43, 65, 91, 121, ...
  33, 35, 37, 39, 41, 63, 89, 119, ...
  51, 53, 55, 57, 59, 61, 87, 117, ...
  73, 75, 77, 79, 81, 83, 85, 115, ...
  ...
The triangular array T(n,k) = M(n-k,k) (with rows n >= 0 and columns k = 0..n) is obtained by reading array M by ascending antidiagonals:
   1;
   3,  7;
   9,  5, 17;
  19, 11, 15, 31;
  33, 21, 13, 29, 49;
  51, 35, 23, 27, 47, 71;
  73, 53, 37, 25, 45, 69, 97;
  99, 75, 55, 39, 43, 67, 95, 127;
  ...

Cf. A001844, A005917, A047926, A056220, A058331.

Antidiagonal sums are in A342362.

tabl(nn) = {my(M=matrix(nn+1,nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n,k]=if(k == n, 2*n^2-2*n+1, if(k < n, 2*n^2-4*n+2*k+1, 2*k^2-2*n+1)))); M}

O.g.f. for rectangular M: (x^4*y^4 + 4*x^3*y^4 + 3*x^4*y^2 - 18*x^3*y^3 - x^2*y^4 + 8*x^3*y^2 + 4*x^2*y^3 - 10*x^3*y + 10*x^2*y^2 - 2*x*y^3 + 8*x^2*y + 4*x*y^2 + 3*x^2 - 18*x*y - y^2 + 4*y + 1)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2).

O.g.f. for triangular T: (x^8*y^4 + 4*x^7*y^4 - x^6*y^4 - 18*x^6*y^3 + 3*x^6*y^2 + 4*x^5*y^3 + 8*x^5*y^2 - 2*x^4*y^3 + 10*x^4*y^2 - 10*x^4*y + 4*x^3*y^2 + 8*x^3*y - x^2*y^2 - 18*x^2*y + 3*x^2 + 4*x*y + 1)/((1 - x)^3*(1 - x*y)^3*(1 - x^2*y)^2).

A067337 Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 9, 1, 4, 8, 14, 27, 1, 5, 12, 22, 41, 81, 1, 6, 17, 34, 63, 122, 243, 1, 7, 23, 51, 97, 185, 365, 729, 1, 8, 30, 74, 148, 282, 550, 1094, 2187, 1, 9, 38, 104, 222, 430, 832, 1644, 3281, 6561, 1, 10, 47, 142, 326, 652, 1262, 2476, 4925, 9842

Offset: 0

Henry Bottomley, Jan 15 2002

			Rows start 1; 1,1; 1,2,3; 1,3,5,9; 1,4,8,14,27; etc. T(4,0)=2*0+1-0=1; T(4,1)=2*1+3-1=4; T(4,2)=2*4+3-3=8; T(4,3)=2*8+1-3=14; T(4,4)=2*14+0-1=27.

Row sums are A025192. Columns include A000012, A000027 and A022856 (essentially). Right hand columns include A000244 (essentially), A007051 and A047926. Central diagonal is A067336.

T(n, k)=2*T(n, k-1)+A037012(n, k). T(n, k)=T(n-1, k-1)+T(n-1, k) if k0.

A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3n-2, 3n-1, 3n, 3n, 3n, 3n.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 36, 36, 36, 36

Offset: 1

Omar E. Pol, Aug 20 2011

Also first differences of A194273 which is also a sequence related to cellular automata.

			Array begins:
1,  2,  3,  3,  3,  3,
4,  5,  6,  6,  6,  6,
7,  8,  9,  9,  9,  9,
10, 11, 12, 12, 12, 12,
13, 14, 15, 15, 15, 15,
16, 17, 18, 18, 18, 18,
19, 20, 21, 21, 21, 21,
22, 23, 24, 24, 24, 24,
...
Sum of row n gives 18*n-3 = A008600(n) - 3.
G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Column 1: A016777. Column 2: A016789. Every column 3, 4, 5 and 6: positive integers of A008585.

Cf. A008600, A047926, A132868, A194273.

[Floor((n+3)/6) + Floor((n+4)/6) + Floor((n+5)/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2015

A194272:=n->floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6): seq(A194272(n), n=1..100); # Wesley Ivan Hurt, Apr 04 2015

Table[Floor[(n + 3)/6] + Floor[(n + 4)/6] + Floor[(n + 5)/6], {n, 100}] (* Wesley Ivan Hurt, Apr 04 2015 *)

x='x+O('x^60); Vec(x*(1-x^3)/((1-x)^2*(1-x^6))) \\ G. C. Greubel, Aug 13 2018

From Michael Somos, May 12 2014: (Start)
Euler transform of length 6 sequence [2, 0, -1, 0, 0, 1].
G.f.: x * (1-x^3) / ( (1-x)^2 * (1-x^6) ).
a(n-1) = A047926(n) - A132868(n). (End)
From Wesley Ivan Hurt, Apr 04 2015, Sep 08 2015: (Start)
a(n) = 2*a(n-1)-a(n-2)-a(n-3)+2*a(n-4)-a(n-5), n>5.
a(n) = floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6).
a(n) = Sum_{i=0..n-1} floor(i/6) - floor((i-3)/6). (End)

A087438 a(n) = 32^(2(n-1)) + 2^(n-2)*(n+1).

Original entry on oeis.org

1, 4, 15, 56, 212, 816, 3184, 12544, 49728, 197888, 789248, 3151872, 12596224, 50360320, 201388032, 805437440, 3221504000, 12885491712, 51540852736, 206161051648, 824639225856, 3298546417664, 13194163650560, 52776608464896

Offset: 0

Paul Barry, Sep 03 2003

Binomial transform of A047926.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

[3*2^(2*(n-1))+2^(n-2)*(n+1): n in [0..25]]; // Vincenzo Librandi, May 21 2011

LinearRecurrence[{8,-20,16},{1,4,15},30] (* or *) CoefficientList[ Series[ (1-4x+3x^2)/((1-2x)^2(1-4x)),{x,0,30}],x] (* Harvey P. Dale, May 20 2011 *)

G.f.: (1 - 4*x + 3*x^2)/((1-2*x)^2*(1-4*x)).
E.g.f.: (3*exp(4*x) + (1+2*x)*exp(2*x))/4.
a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3); a(0)=1, a(1)=4, a(2)=15. - Harvey P. Dale, May 20 2011

A099265 Partial sums of A056272.

Original entry on oeis.org

1, 3, 8, 23, 75, 277, 1132, 4977, 22979, 109451, 531456, 2610931, 12917683, 64181625, 319695980, 1594859885, 7963472187, 39784944799, 198827606704, 993846943839, 4968361974491, 24839192686973, 124188113975628, 620917025694793, 3104514504312595, 15522360665856147, 77611167795714752

Offset: 1

Nelma Moreira, Oct 10 2004

Density of regular language L{0}* over {0, 1, 2, 3, 4, 5} (i.e., the number of strings of length n), where L is described by regular expression with c = 5: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + 11*2(1 + 2)* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*)0*).

Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

Cf. A008277, A047926, A056272, A099264, A099266.

with (combinat):seq(sum(sum(stirling2(k, j),j=1..5), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007

a(n) = sum(m=1, n, sum(i=1, 5, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 10 2021

a(5,n) = (1/96)*5^n + (1/8)*3^n + (1/3)*2^n + (3/8)*n - 15/32.
a(n) = Sum_{m=1..n} Sum_{i=1..5} S(m,i), where S(m,i) = A008277(m,i) (i.e., partial sum of the sum of Stirling numbers of second kind S(n,i) for i = 1..5).
For c = 5, a(c,n) = g(1,c)*n + Sum_{k=2..c} g(k,c)*k*(k^n - 1)/(k - 1), where g(1,1) = 1, g(1,c) = g(1,c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k,c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: x*(-1 + 19*x^3 - 24*x^2 + 9*x)/((3*x-1)*(2*x-1)*(5*x-1)*(x-1)^2). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]

Name and Formula section edited by Petros Hadjicostas, Mar 10 2021

More terms from Michel Marcus, Jan 05 2025

A099266 Partial sums of A056273.

Original entry on oeis.org

1, 3, 8, 23, 75, 278, 1154, 5265, 25913, 135212, 736704, 4139831, 23767895, 138468210, 814675838, 4824766301, 28699128501, 171207852152, 1023332115836, 6124430348355, 36684624841811, 219860794899518, 1318179574171578

Offset: 1

Nelma Moreira, Oct 10 2004

Some previous names were a(6,n) := (1/600)*6^n + (1/36)*4^n + (1/12)*3^n + (3/8)*2^n + (11/30)*n - (439/900) = Sum_{m=1..n} Sum_{i=1..6} S(m,i), where S(n,i) = A008277(n,i) are the Stirling numbers of the second kind.

Density of the regular language L{0}* over {0, 1, 2, 3, 4, 5, 6} (i.e., the number of strings of length n), where L is described by regular expression with c = 6: Sum_{i=1..c} Prod_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*6(1 + 2 + 3 + 4 + 5 + 6)*)0*).

Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (17,-111,355,-584,468,-144).

Cf. A047926, A056273, A008277, A099264, A099265.

with (combinat):seq(sum(sum(stirling2(k, j),j=1..6), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007

Vec(x*(91*x^4-135*x^3+68*x^2-14*x+1)/((x-1)^2*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Oct 28 2014

a(n) = sum(m=1, n, sum(i=1, 6, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 09 2021

For c = 6, a(c, n) = g(1, c)*n + Sum_{k=2..c} g(k, c)*k*(k^n - 1)/(k - 1), where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.

G.f.: x*(91*x^4 - 135*x^3 + 68*x^2 - 14*x + 1) / ((x - 1)^2*(2*x - 1)*(3*x - 1)*(4*x - 1)*(6*x - 1)). - Colin Barker, Oct 28 2014

Shorter name by Joerg Arndt, Oct 28 2014

Comments edited by Petros Hadjicostas, Mar 09 2021

cp's OEIS Frontend

A094176 (A047926(n)-A091588(n))/2.

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A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2, read by antidiagonals.

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A342354 M(n,k) = 2*n^2 + 2*k + 1 for 0 <= k <= n and M(n,k) = 2*k^2 + 4*k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

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A087438 a(n) = 3*2^(2*(n-1)) + 2^(n-2)*(n+1).

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