A052161 -id:A052161 - cp's OEIS frontend

A014825 a(n) = 4*a(n-1) + n with n > 1, a(1)=1.

1, 6, 27, 112, 453, 1818, 7279, 29124, 116505, 466030, 1864131, 7456536, 29826157, 119304642, 477218583, 1908874348, 7635497409, 30541989654, 122167958635, 488671834560, 1954687338261, 7818749353066

Offset: 1

N. J. A. Sloane

			G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
Hacène Belbachir and El-Mehdi Mehiri, Enumerating moves in the optimal solution of the Tower of Hanoi, arXiv:2210.08657 [math.CO], 2022.
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A002450 (first differences), A052161 (partial sums).
Cf. A001045, A006314, A014916, A053142, A091919.
Cf. A171654 (mod 10).

[(4^(n+1)-3*n-4)/9: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

RecurrenceTable[{a[1]==1,a[n]==4a[n-1]+n},a[n],{n,30}] (* Harvey P. Dale, Oct 12 2011 *)
a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* Michael Somos, Jun 20 2012 *)

{a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */

def A014825(n): return (((1<<(n+1<<1))-4)//3-n)//3 # Chai Wah Wu, Nov 12 2024

[(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020

a(n) = (4^(n+1) - 3*n - 4)/9.
G.f.: x/((1-4*x)*(1-x)^2).
a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - Paul Barry, Oct 23 2009
Convolution square of A006314. - Michael Somos, Jun 20 2012
E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - G. C. Greubel, Feb 18 2020
a(n) = A014916(-n-1)*4^(n+1) = A091919(2*n-2) for all n in Z. - Michael Somos, Oct 02 2020
a(n) = Sum_{k=0..n} A002450(k). - Joseph Brown, May 11 2021
Last digits give A171654. - Paul Curtz, Oct 10 2021

A097788 a(n)=4a(n-1)+C(n+3,3),n>0, a(0)=1.

Original entry on oeis.org

1, 8, 42, 188, 787, 3204, 12900, 51720, 207045, 828400, 3313886, 13255908, 53024087, 212096908, 848388312, 3393554064, 13574217225, 54296870040, 217187481490, 868749927500, 3474999711771, 13899998849108, 55599995398732

Offset: 0

Paul Barry, Aug 24 2004

Partial sums of A052161.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-22,28,-17,4).

RecurrenceTable[{a[0]==1,a[n]==4a[n-1]+Binomial[n+3,3]},a,{n,30}] (* or *) LinearRecurrence[{8,-22,28,-17,4},{1,8,42,188,787},30] (* Harvey P. Dale, May 04 2014 *)

G.f.: 1/((1-4x)(1-x)^4); a(n)=4^(n+4)/81-(9n^3+90n^2+303n+350)/162; a(n)=sum{k=0..n, binomial(n+4, k+4)3^k}.

a(0)=1, a(1)=8, a(2)=42, a(3)=188, a(4)=787, a(n)=8*a(n-1)- 22*a(n-2)+ 28*a(n-3)- 17*a(n-4)+4*a(n-5). - Harvey P. Dale, May 04 2014

A262592 a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8.

Original entry on oeis.org

1, 2, 4, 10, 29, 88, 268, 812, 2449, 7366, 22124, 66406, 199261, 597836, 1793572, 5380792, 16142465, 48427498, 145282612, 435847970, 1307544061, 3922632352, 11767897244, 35303691940, 105911076049, 317733228398, 953199685468, 2859599056702, 8578797170429, 25736391511636

Offset: 0

N. J. A. Sloane, Oct 21 2015

Colin Barker, Table of n, a(n) for n = 0..1000
K. Satyanarayana, Sequences whose kth differences form a geometrical progression, Math. Student, 12 (1944), page 109. [Annotated scanned copy. This sequence was formerly A2752 but has now been renumbered]
Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Other sequences with generating functions like this: A000340, A052161, A262593, A262594.

f1:=(a,b)->(1-a*x)^a/((1-x)^b*(1-b*x));
f2:=(a,b)->seriestolist(series(f1(a,b),x,40));
f2(2,3);

Table[3^(n + 1)/8 + 5/8 - n^2/4 + n/2, {n, 0, 29}] (* Michael De Vlieger, Oct 23 2015 *)
LinearRecurrence[{6,-12,10,-3},{1,2,4,10},30] (* Harvey P. Dale, Jul 18 2025 *)

a(n) = 3^(n+1)/8+5/8-n^2/4+n/2 \\ Colin Barker, Oct 23 2015

Vec((1-2*x)^2/((1-x)^3*(1-3*x)) + O(x^40)) \\ Colin Barker, Oct 23 2015

G.f.: (1-2*x)^2/((1-x)^3*(1-3*x)).

a(n) = 6*a(n-1)-12*a(n-2)+10*a(n-3)-3*a(n-4) for n>3. - Colin Barker, Oct 23 2015

A326294 Number of connected simple graphs on a subset of {1..n} with no crossing or nesting edges.

Original entry on oeis.org

1, 1, 2, 8, 35, 147, 600, 2418

Offset: 0

Gus Wiseman, Jun 29 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

			The a(4) = 35 edge-sets:
  {}  {12}  {12,13}  {12,13,14}  {12,13,14,34}
      {13}  {12,14}  {12,13,23}  {12,13,23,34}
      {14}  {12,23}  {12,13,34}  {12,14,24,34}
      {23}  {12,24}  {12,14,24}  {12,23,24,34}
      {24}  {13,14}  {12,14,34}
      {34}  {13,23}  {12,23,24}
            {13,34}  {12,23,34}
            {14,24}  {12,24,34}
            {14,34}  {13,14,34}
            {23,24}  {13,23,34}
            {23,34}  {14,24,34}
            {24,34}  {23,24,34}

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

The inverse binomial transform is the covering case A326339.
Covering graphs with no crossing or nesting edges are A326329.
Connected simple graphs are A001349.
Graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326340.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

Conjecture: a(n) = A052161(n - 2) + 1.

A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

Original entry on oeis.org

0, 1, 8, 41, 180, 745, 3016, 12113, 48516, 194145, 776680, 3106841, 12427508, 49710201, 198841000, 795364225, 3181457156, 12725828913, 50903315976, 203613264265, 814453057460, 3257812230281, 13031248921608, 52124995686961, 208499982748420, 833999930994305

Offset: 0

Seiichi Manyama, Dec 28 2023

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-15,13,-4).

Cf. A000290, A000330, A047520, A368528.
Cf. A052161, A368524.
Cf. A014825, A368530.

LinearRecurrence[{7, -15, 13, -4}, {0, 1, 8, 41}, 30] (* Paolo Xausa, Jan 29 2024 *)

PARI
```
a(n) = sum(k=1, n, k^2*4^(n-k));
```

G.f.: x * (1+x)/((1-4*x) * (1-x)^3).
a(n) = 7*a(n-1) - 15*a(n-2) + 13*a(n-3) - 4*a(n-4).
a(n) = A052161(n-1) + A052161(n-2) for n > 1.
a(n) = (5*4^(n+1) - (9*n^2 + 24*n + 20))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^2.

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

Original entry on oeis.org

0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A062805, A368535.
Cf. A002662, A052150, A052161.
Cf. A023037, A062806, A068475.

Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)

a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));

a(n) = [x^n] x/((1-n*x) * (1-x)^3).

a(n) = n * (2*n^(n+1) - n^3 - n^2 + n - 1)/(2 * (n-1)^3) for n > 1.

A229611 Expansion of 1/((1-x)^3*(1-11x)).

Original entry on oeis.org

1, 14, 160, 1770, 19485, 214356, 2357944, 25937420, 285311665, 3138428370, 34522712136, 379749833574, 4177248169405, 45949729863560, 505447028499280, 5559917313492216, 61159090448414529, 672749994932559990, 7400249944258160080, 81402749386839761090

Offset: 0

Yahia Kahloune, Sep 26 2013

This sequence was chosen to illustrate a method of matching generating functions and closed-form solutions: The general term associated with the generating function 1/((1-s*x)^3*(1-r*x)) with r>s>=1 is a(n) = [r^(n+3) - s^(n+1)*(s^2 + (r-s)*s*binomial(n+3,1) +(r-s)^2*binomial(n+3,2))] / (r-s)^3 .

			a(3) = (11^6 - (50*3^2+260*3 + 331))/1000 = 1770 .

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-36,34,-11).

Cf. A002662, A052150, A052161, A052244, A052262.

[(11^(n+3) - (50*n^2 + 260*n + 331))/1000: n in [0..25]]; // Vincenzo Librandi, Sep 27 2013

CoefficientList[Series[1/((1 - x)^3 (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 27 2013 *)
LinearRecurrence[{14,-36,34,-11},{1,14,160,1770},30] (* Harvey P. Dale, Apr 09 2016 *)

a(n) = (11^(n+3) - (1 + 10*C(n+3,1) + 100*C(n+3,2)))/1000 = (11^(n+3) - (50*n^2 + 260*n + 331))/1000.

a(n) = 14*a(n-1) -36*a(n-2) +34*a(n-3) -11*a(n-4). - Vincenzo Librandi, Sep 27 2013

cp's OEIS Frontend

A014825 a(n) = 4*a(n-1) + n with n > 1, a(1)=1.

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A097788 a(n)=4a(n-1)+C(n+3,3),n>0, a(0)=1.

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A262592 a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8.

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A326294 Number of connected simple graphs on a subset of {1..n} with no crossing or nesting edges.

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A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

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A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

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A229611 Expansion of 1/((1-x)^3*(1-11x)).

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Magma

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