A015441 -id:A015441 - cp's OEIS frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A053455 a(n) = ((8^n) - (-6)^n)/14.

Original entry on oeis.org

0, 1, 2, 52, 200, 2896, 15392, 169792, 1078400, 10306816, 72376832, 639480832, 4753049600, 40201179136, 308548739072, 2546754076672, 19903847628800, 162051890937856, 1279488468058112, 10337467701133312, 82090381869056000, 660379213392510976, 5261096756499709952, 42220395755839946752

Offset: 0

Barry E. Williams, Jan 13 2000

Previous name was: A linear recursive sequence.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,48).

Cf. A053404, A051958, A015441, A080921.

A053455:=n->((8^n)-(-6)^n)/14; seq(A053455(n), n=0..30); # Wesley Ivan Hurt, Mar 07 2014

LinearRecurrence[{2,48},{1,2},30] (* Harvey P. Dale, Nov 28 2011 *)
CoefficientList[Series[x / (1 - 2 x - 48 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Mar 08 2014 *)

a(n) = ((8^n)-(-6)^n)/14; \\ Joerg Arndt, Mar 08 2014

a(n) = 2*a(n-1) + 48*a(n-2), n>=2; a(0)=0, a(1)=1.
a(n) = ((8^n)-(-6)^n)/14 = (2^(n-1))*((4^n) - (-3)^n)/7 = 2^(n-1)*A053404(n).
G.f.: x/((1+6*x)*(1-8*x)). - Harvey P. Dale, Nov 28 2011
a(n) = A080921(n). - Philippe Deléham, Mar 05 2014
a(n+1) = Sum_{k=0..n} A238801(n,k)*7^k. - Philippe Deléham, Mar 07 2014

More terms from James Sellers, Feb 02 2000

New name (from formula), Joerg Arndt, Mar 05 2014

A060959 Table by antidiagonals of generalized Fibonacci numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=0 and T(n,1)=1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 5, 5, 4, 1, 1, 0, 1, 8, 11, 7, 5, 1, 1, 0, 1, 13, 21, 19, 9, 6, 1, 1, 0, 1, 21, 43, 40, 29, 11, 7, 1, 1, 0, 1, 34, 85, 97, 65, 41, 13, 8, 1, 1, 0, 1, 55, 171, 217, 181, 96, 55, 15, 9, 1, 1, 0, 1, 89, 341, 508, 441, 301, 133, 71, 17, 10, 1, 1, 0

Offset: 0

Henry Bottomley, May 10 2001

			Square array begins as:
  0, 1, 1, 1,  1,  1,  1, ...
  0, 1, 1, 2,  3,  5,  8, ...
  0, 1, 1, 3,  5, 11, 21, ...
  0, 1, 1, 4,  7, 19, 40, ...
  0, 1, 1, 5,  9, 29, 65, ...
  0, 1, 1, 6, 11, 41, 96, ...

G. C. Greubel, Antidiagonals n = 0..100, flattened

Rows include A057427, A000045, A001045, A006130, A006131, A015440, A015441, A015442, A015443, A015445, A015446, A015447, A053404.

Columns include A000004, A000012 (twice), A000027, A000567 A005408, A028387.

Flat(List([0..12], n-> List([0..n], k-> (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k))/Sqrt(1+4*k) ))); # G. C. Greubel, Jan 15 2020

[Round( (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k) )/Sqrt(1+4*k) ): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 15 2020

seq(seq( round((((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k)), k=0..n), n=0..12); # G. C. Greubel, Jan 15 2020

T[n_, k_]:= If[n==k==0, 0, Round[(((1+Sqrt[1+4n])/2)^k - ((1-Sqrt[1+4n])/2)^k)/Sqrt[1+4n]]]; Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 15 2020 *)

T(n,k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n);
for(n=0,12, for(k=0,n, print1( round(T(k,n-k)), ", "))) \\ G. C. Greubel, Jan 15 2020

[[ round( (((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 15 2020

T(n, k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n).

A080920 a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 39, 148, 1661, 8502, 75139, 447848, 3525561, 22725802, 168846239, 1133095548, 8175809461, 56009963102, 398173257339, 2756695223248, 19449454453361, 135383241720402, 951497389308439, 6641408238830948

Offset: 0

Paul Barry, Feb 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,35).

Cf. A079773, A051958, A015441.

Join[{a=0,b=1},Table[c=2*b+35*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1 / ((1 + 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,35},{0,1},30] (* Harvey P. Dale, Aug 24 2017 *)

a(n) = 7^n/12 - (-5)^n/12.
a(n) = Sum{k=1..n, binomial(n, 2k-1)*6^(2(k-1))}.
G.f.: 1/((1+5x)(1-7x)).
a(n+1) = Sum_{k = 0..n} A238801(n,k)*6^k. - Philippe Deléham, Mar 07 2014

A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.

Original entry on oeis.org

1, 1, 43, 85, 1891, 5461, 84883, 314245, 3879331, 17077621, 180009523, 897269605, 8457669571, 46142992981, 401365114963, 2339370820165, 19196705648611, 117450280095541, 923711917337203, 5856623681349925, 44652524209512451, 290630718826209301, 2166036735625732243

Offset: 0

Len Smiley, Dec 07 2001

A second-order recurrence of promic type (integer roots).
If the number j = A002378(m) is promic (= i(i+1)), then a(n) = a(n-1) + j*a(n-2), a(0) = a(1) = 1 has a closed-form solution involving only powers of integers. The binomial coefficient sum solves the recurrence regardless of promicity (cf. GKP reference).
Hankel transform is := 1,42,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, p. 204.

Harry J. Smith, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (1,42).

Cf. A001045 (j=2), A015441 (j=6), A053404 (j=12), A053428 (j=20), A053430 (j=30).

Maple
```
n->sum(binomial(n-k, k)*(42)^k, k=0..n)
```

LinearRecurrence[{1,42},{1,1},30] (* Harvey P. Dale, Apr 30 2017 *)

a(n) = { (7^(n+1) - (-6)^(n+1))/13 } \\ Harry J. Smith, Nov 02 2009

a(n) = a(n-1) + 42a(n-2); a(0) = a(1) = 1.

G.f.: -1/((6*x+1)*(7*x-1)). - R. J. Mathar, Nov 16 2007

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 52, 200, 2896, 15392, 169792, 1078400, 10306816, 72376832, 639480832, 4753049600, 40201179136, 308548739072, 2546754076672, 19903847628800, 162051890937856, 1279488468058112, 10337467701133312, 82090381869056000

Offset: 0

Paul Barry, Feb 24 2003

Essentially the same as A053455: a(n) = A053455(n-1), n>=1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,48).

Cf. A079773, A051958, A015441, A080920.

CoefficientList[Series[x / ((1 + 6 x) (1 - 8 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,48},{0,1},30] (* Harvey P. Dale, Jan 20 2016 *)

a(n) = (8^n - (-6)^n)/14.
a(n) = Sum{k=1..n, binomial(n, 2k-1) * 7^(2(k-1)) }
G.f.: x/((1+6*x)*(1-8*x)).
a(n) = A053455(n-1), n>=1. [R. J. Mathar, Sep 18 2008]

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

cp's OEIS Frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

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A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A053455 a(n) = ((8^n) - (-6)^n)/14.

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A060959 Table by antidiagonals of generalized Fibonacci numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=0 and T(n,1)=1.

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A080920 a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.

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A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.