A006234 -id:A006234 - cp's OEIS frontend

A027261 a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).

1, 4, 18, 72, 270, 972, 3402, 11664, 39366, 131220, 433026, 1417176, 4605822, 14880348, 47829690, 153055008, 487862838, 1549681956, 4907326194, 15496819560, 48814981614, 153418513644, 481176247338, 1506290861232, 4707158941350, 14686335897012, 45753584909922

Offset: 0

Clark Kimberling

nonn

Cf. A014915, A064017, A079272.

Cf. A006234.

a(n) = 2(n+1)*3^(n-1), for n>1 (conjectured). - Ralf Stephan, Feb 02 2004
From Colin Barker, Jul 28 2012: (Start)
Conjecture: a(n) = 6*a(n-1)-9*a(n-2), for n>3.
G.f.: (1-9*x^2+18*x^3)/(1-3*x)^2. (End)

a(1) corrected and more terms from Sean A. Irvine, Oct 26 2019

A081106 6th binomial transform of (1,1,0,0,0,0,...).

Original entry on oeis.org

1, 7, 48, 324, 2160, 14256, 93312, 606528, 3919104, 25194240, 161243136, 1027924992, 6530347008, 41358864384, 261213880320, 1645647446016, 10344069660672, 64885527871488, 406239826673664, 2538998916710400, 15843353240272896, 98716277881700352, 614234617930579968

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 5*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A079028, A081105, A006234.

[(n+6)*6^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 5 x)/(1 - 6 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{12,-36},{1,7},30] (* Harvey P. Dale, Nov 07 2013 *)

a(n) = 12*a(n-1) - 36*a(n-2) with n > 1, a(0) = 1, a(1) = 7.
a(n) = (n + 6)*6^(n-1).
G.f.: (1 - 5*x)/(1 - 6*x)^2.
E.g.f.: exp(6*x)*(1 + x). - Stefano Spezia, Mar 05 2023

A081108 8th binomial transform of (1,1,0,0,0,0,...).

Original entry on oeis.org

1, 9, 80, 704, 6144, 53248, 458752, 3932160, 33554432, 285212672, 2415919104, 20401094656, 171798691840, 1443109011456, 12094627905536, 101155069755392, 844424930131968, 7036874417766400, 58546795155816448, 486388759756013568

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(0,j) = j; m(i,0) = i and m(i,j) = m(i-1,j) + 7*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A081107, A081109, A006234.

[(n+8)*8^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

LinearRecurrence[{16,-64},{1,9},30] (* Harvey P. Dale, Jun 11 2013 *)
CoefficientList[Series[(1 - 7 x) / (1 - 8 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n)=(n+8)*8^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*a(n-1) - 64*a(n-2), a(0) = 1, a(1) = 9.
a(n) = (n + 8)*8^(n-1).
G.f.: (1 - 7*x)/(1 - 8*x)^2.
E.g.f.: exp(8*x)*(1 + x). - Stefano Spezia, Mar 04 2023

A081279 Binomial transform of Chebyshev coefficients A001794.

Original entry on oeis.org

1, 8, 47, 238, 1101, 4788, 19899, 79866, 311769, 1189728, 4454919, 16415622, 59659173, 214229772, 761200659, 2679525522, 9353893041, 32409397944, 111534054111, 381480041502, 1297471217661, 4390248981348, 14785128121707

Offset: 0

Paul Barry, Mar 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A007051, A006234, A081278, A081280.

[(2*n^3+30*n^2 + 103*n + 81)*3^(n - 4): n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[(1 - 2 x) (1 - x)^2 / (1 - 3 x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-54,108,-81},{1,8,47,238},30] (* Harvey P. Dale, Jul 27 2015 *)

a(n) = (2*n^3 + 30*n^2 + 103*n + 81) * 3^(n-4).
a(n) = 12*a(n-1) -54*a(n-2) +108*a(n-3) +8*1a(n-4), a(0)=1, a(1)=8, a(2)=47, a(3)=238.
G.f.: (1-2*x)*(1-x)^2/(1-3*x)^4.

A103247 Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.

Original entry on oeis.org

1, -3, 1, 8, -6, 1, -20, 24, -9, 1, 48, -80, 48, -12, 1, -112, 240, -200, 80, -15, 1, 256, -672, 720, -400, 120, -18, 1, -576, 1792, -2352, 1680, -700, 168, -21, 1, 1280, -4608, 7168, -6272, 3360, -1120, 224, -24, 1, -2816, 11520, -20736, 21504, -14112, 6048, -1680, 288, -27, 1, 6144, -28160, 57600, -69120, 53760, -28224, 10080, -2400, 360, -30, 1

Offset: 0

Emeric Deutsch, Mar 19 2005

Row sums of the unsigned triangle yield A006234. The unsigned triangle is the mirror image of A103407.

			The monic characteristic polynomial of the matrix [3 1 1 / 1 3 1 / 1 1 3] is x^3 - 9x^2 + 24x - 20; so T(3,0)=-20, T(3,1)=24, T(3,2)=-9, T(3,3)=1.
Triangle begins:
  1;
  -3,1;
  8,-6,1;
  -20,24,-9,1;
  48,-80,48,-12,1;
  ...

Cf. A006234, A103407.

with(linalg): a:=proc(i,j) if i=j then 3 else 1 fi end: 1;for n from 1 to 10 do seq(coeff(expand(x*charpoly(matrix(n,n,a),x)),x^k),k=1..n+1) od; # yields the sequence in triangular form

M[n_] := Table[If[i == j, 3, 1], {i, 1, n}, {j, 1, n}];
P[n_] := P[n] = CharacteristicPolynomial[M[n], x];
row[n_] := row[n] = If[n == 0, {1}, CoefficientList[P[n]/Coefficient[P[n], x, n], x]];
T[n_, k_] := row[n][[k]];
Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 06 2024 *)

Appears to be the matrix product (I-S)*P^(-2), where I is the identity, P is Pascal's triangle A007318 and S is A132440, the infinitesimal generator of P. Cf. A055137 (= (I-S)*P) and A103283 (= (I-S)*P^(-1)). - Peter Bala, Nov 28 2011

A081105 5th binomial transform of (1,1,0,0,0,0,.....).

Original entry on oeis.org

1, 6, 35, 200, 1125, 6250, 34375, 187500, 1015625, 5468750, 29296875, 156250000, 830078125, 4394531250, 23193359375, 122070312500, 640869140625, 3356933593750, 17547607421875, 91552734375000, 476837158203125

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(1,j)=j; m(i,1)=i and m(i,j)=m(i-1,j)+4*m(i-1,j-1) - Benoit Cloitre, Jun 13 2003

Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A001792, A006234, A079027, A092288.

LinearRecurrence[{10,-25},{1,6},30] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=if(n,([0,1; -25,10]^n*[1;6])[1,1],1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*a(n-1)-25*a(n-2), a(0)=1, a(1)=6.
a(n) = (n+5)*5^(n-1).
G.f.: (1-4x)/(1-5x)^2.
a(n) = A079027(n), n>0. - R. J. Mathar, Sep 18 2008
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 15625*log(5/4) - 41825/12.
Sum_{n>=0} (-1)^n/a(n) = 15625*log(6/5) - 34175/12. (End)

A081109 9th binomial transform of (1,1,0,0,0,0,0,...).

Original entry on oeis.org

1, 10, 99, 972, 9477, 91854, 885735, 8503056, 81310473, 774840978, 7360989291, 69735688020, 659002251789, 6213449802582, 58462914051567, 549043018919064, 5147278302366225, 48178524910147866, 450283905890997363

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(0,j) = j; m(i,0) = i and m(i,j) = m(i-1,j) + 8*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (18, -81).

Cf. A081108, A081122, A006234.

[(n+9)*9^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 8 x) / (1 - 9 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n) = 18*a(n-1) - 81*a(n-2), a(0) = 1, a(1) = 10.
a(n) = (n + 9)*9^(n-1).
G.f.: (1 - 8*x)/(1 - 9*x)^2.
E.g.f.: exp(9*x)*(1 + x). - Stefano Spezia, Mar 04 2023

A081909 a(n) = 3^n(n^2 - n + 18)/18.

Original entry on oeis.org

1, 3, 10, 36, 135, 513, 1944, 7290, 26973, 98415, 354294, 1259712, 4428675, 15411789, 53144100, 181752822, 617003001, 2080591515, 6973568802, 23245229340, 77096677311, 254535261273, 836828256240, 2740612539186

Offset: 0

Paul Barry, Mar 31 2003

Binomial transform of A081908. 3rd binomial transform of (1,0,1,0,0,0,...). Case k=3 where a(n,k) = k^n*(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.

a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used exactly twice. - Enrique Navarrete, Mar 29 2024

			a(2)=10 since the number of words of length 2 defined on {0,1,2,3} that don't use 0 or use it twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 00. - _Enrique Navarrete_, Mar 29 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..185
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A006234, A081910.

[3^n*(n^2-n+18)/18: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

Table[3^n(n^2-n+18)/18,{n,0,30}] (* or *) CoefficientList[Series[ (1-6x+10x^2)/(1-3x)^3,{x,0,30}],x]  (* Harvey P. Dale, Apr 26 2011 *)

a(n) = 3^n*(n^2 - n + 18)/18.
G.f.: (1 - 6x + 10x^2)/(1-3x)^3.
E.g.f.: exp(3*x)*(1+x^2/2). - Enrique Navarrete, Mar 29 2024

A081107 7th binomial transform of (1,1,0,0,0,0,...).

Original entry on oeis.org

1, 8, 63, 490, 3773, 28812, 218491, 1647086, 12353145, 92236816, 686011319, 5084554482, 37569208117, 276825744020, 2034669218547, 14920907602678, 109193914728689, 797590333670424, 5815762849680175, 42338753545671674, 307770170005074861, 2234183456333136028

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(0,j) = j; m(i,0) = i and m(i,j) = m(i-1,j) + 6*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A081106, A081108, A006234.

[(n+7)*7^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 6 x)/(1 - 7 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n) = 14*a(n-1) - 49*a(n-2) with n > 1, a(0) = 1, a(1) = 8.
a(n) = (n + 7)*7^(n-1).
G.f.: (1 - 6*x)/(1 - 7*x)^2.
E.g.f.: exp(7*x)*(1 + x). - Stefano Spezia, Mar 05 2023

A081278 Binomial transform of Chebyshev polynomial coefficients A001793.

Original entry on oeis.org

1, 6, 29, 126, 513, 1998, 7533, 27702, 99873, 354294, 1240029, 4290894, 14703201, 49955454, 168466797, 564390342, 1879706817, 6227425638, 20533285917, 67411165086, 220442258241, 718277586606, 2332658764269, 7552375012566

Offset: 0

Paul Barry, Mar 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A007051, A006234, A081279.

[(n^2+8*n+9)*3^(n-2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[(1 - 2 x) (1 - x) / (1 - 3 x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)

a(n) = (n^2+8*n+9)3^(n-2).

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

cp's OEIS Frontend

A027261 a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).

Views

Author

Keywords

Crossrefs

Formula

Extensions

A081106 6th binomial transform of (1,1,0,0,0,0,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A081108 8th binomial transform of (1,1,0,0,0,0,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A081279 Binomial transform of Chebyshev coefficients A001794.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A103247 Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A081105 5th binomial transform of (1,1,0,0,0,0,.....).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A081109 9th binomial transform of (1,1,0,0,0,0,0,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A081909 a(n) = 3^n(n^2 - n + 18)/18.

Views

Author

Keywords

Comments

Examples