A110523 -id:A110523 - cp's OEIS frontend

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

N. J. A. Sloane, May 03 2000

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

More terms from Michel Marcus, Jul 25 2023

A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, -1, -2, 5, 1, -16, 13, 35, -74, -31, 253, -160, -599, 1079, 718, -3955, 1801, 10064, -15467, -14725, 61126, -16951, -166427, 217280, 282001, -933841, 87838, 2713685, -2977199, -5163856, 14095453, 1396115, -43682474, 39494129, 91553293, -210035680

Offset: 0

Roman Witula, Jul 27 2012

The sequence a(n) is conjugate with A110523 by the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 + i*sqrt(11))/2, or ((-1 - i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 - i*sqrt(11))/2 (see also comments to A110523, where these relations and many other facts on a(n) is presented).
Apart from signs, the Lucas U(P=1,Q=3)-sequence. - R. J. Mathar, Oct 24 2012
This is the Lucas U(-1, 3) sequence. (V_n(-1, 3))^2 + 11*(U_n(-1, 3))^2 = 4*Q^n = 4*3^n. For the special case where |U_n(-1, 3)| = 1, then, by the Lucas sequence identity U_2*n = U_n*V_n, we have (U_2*n(-1, 3))^2 + 11 = 4*3^n, true for n = 1, 2, 5, U_n = 1, -1, 1 and U_2*n = -1, 5, -31. E.g., (-31)^2 + 11 = 972 = 4*3^5. - Raphie Frank, Dec 09 2015

Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.

Seiichi Manyama, Table of n, a(n) for n = 0..4189
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (-1,-3).

Cf. A106852, A110523.

[n le 2 select n-1 else -Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 10 2015

LinearRecurrence[{-1, -3}, {0, 1}, 40] (* T. D. Noe, Jul 30 2012 *)

concat(0,Vec(1/(1+x+3*x^2)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

[(-1)^(n-1)*round(3^((n-1)/2)*chebyshev_U(n-1, 1/(2*sqrt(3)))) for n in range(41)] # G. C. Greubel, Dec 28 2023

a(n) = - a(n-1) - 3*a(n-2).
a(n) = (-1)^n*(i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^n - ((1 - i*sqrt(11))/2)^n).
G.f.: x/(1 + x + 3*x^2).
G.f.: Q(0) -1, where Q(k) = 1 - 3*x^2 - (k+2)*x + x*(k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
From G. C. Greubel, Dec 28 2023: (Start)
a(n) = (-1)^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, 1/(2*sqrt(3))).
a(n) = (-1)^n * A106852(n-1).
E.g.f.: (2/sqrt(11))*exp(-x/2)*sin(sqrt(11)*x/2). (End)

A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A003688, A003945.

With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
From Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - Philippe Deléham, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 31 2013

A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2).

Original entry on oeis.org

1, -1, 1, 1, -5, 1, -1, 12, -9, 1, 1, -22, 39, -13, 1, -1, 35, -115, 82, -17, 1, 1, -51, 270, -344, 141, -21, 1, -1, 70, -546, 1106, -773, 216, -25, 1, 1, -92, 994, -2954, 3199, -1466, 307, -29, 1, -1, 117, -1674, 6888, -10791, 7461, -2487, 414, -33, 1, 1, -145, 2655, -14484, 31179, -30645, 15060, -3900, 537, -37, 1

Offset: 0

Paul Barry, Jul 24 2005

Inverse of A110519.

Product of inverse binomial transform matrix (1/(1+x), x/(1+x)) and (1, x*(1-3*x)) (A110517).

			Rows begin
   1;
  -1,    1;
   1,   -5,    1;
  -1,   12,   -9,    1;
   1,  -22,   39,  -13,    1;
  -1,   35, -115,   82,  -17,    1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, J. Int. Seq. 16 (2013) #13.5.4.

Cf. A110519 (inverse), A110523 (row sums), A110524 (diagonal sums).

Cf. A000326, A016813, A033999, A052924, A078005, A110517, A236267.

A110522:= func< n,k | (-1)^(n+k)*(&+[ 3^(j-k)*Binomial(k,j-k)*Binomial(n,j) : j in [0..n]] ) >;
[A110522(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 28 2023

T[n_,k_]:= Sum[(-1)^(n-j)*(-3)^(j-k)*Binomial[k, j- k]*Binomial[n, j], {j,0,n}];
Table[T[n,k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 30 2017 *)

A110522(n,k) = if(n==0, 1, sum(j=0,n, (-1)^(n-j)*(-3)^(j-k)*binomial(n,j)*binomial(k, j-k)));
for(n=0,12, for(k=0,n, print1(A110522(n,k), ", "))) \\ G. C. Greubel, Aug 30 2017; Dec 28 2023

def A110522(n,k): return (-1)^(n+k)*sum(3^(j-k)*binomial(k,j-k)*binomial(n,j) for j in range(n+1))
flatten([[A110522(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 28 2023

Number triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k)*2^i.
Sum_{k=0..n} T(n, k) = A110523(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A110524(n) (diagonal sums).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(1,0) = -1, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014
From G. C. Greubel, Dec 28 2023: (Start)
T(n, 0) = A033999(n).
T(n, 1) = (-1)^(n-1)*A000326(n), n >= 1.
T(n, n) = 1.
T(n, n-1) = -A016813(n-1), n >= 1.
T(n, n-2) = A236267(n-2), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A052924(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A078005(n). (End)

A103646 G.f.: 9*(3x+1)/(1+2x-6x^2-27x^3).

Original entry on oeis.org

9, 9, 36, 225, 9, 2304, 1521, 11025, 49284, 8649, 576081, 230400, 3229209, 10478169, 4639716, 140778225, 29192409, 911556864, 2153052801, 1951430625, 33627490884, 2586027609, 249281516961, 424895385600, 715721076009, 7848531119529

Offset: 0

Creighton Dement, Feb 12 2005

A floretion-generated sequence of squares. This sequence is related to several other sequences of squares.

Floretion Algebra Multiplication Program, FAMP Code: 4ibaseiseq[ x*(+ 'i + 'j + i' + j' + 'ii' + 'jj' + 'ij' + 'ji' + e) ] where x is the sum of all (16) floretion basis vectors.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,6,27).

Cf. A103644, A103645.

CoefficientList[ Series[9(3x + 1)/(1 + 2x - 6x^2 - 27x^3), {x, 0, 25}], x] (* Robert G. Wilson v, Feb 12 2005 *)
LinearRecurrence[{-2,6,27},{9,9,36},40] (* Harvey P. Dale, Mar 14 2016 *)

a(n+3) = -2a(n+2) + 6a(n+1) + 27a(n), a(0) = 9, a(1) = 9, a(2) = 36
a(n) = 9*A103644(n).
4*3^(n+1) = 2*A103644(n) + a(n) + A103645(n) = 11*A103644(n) + A103645(n).
a(n) = A110523(n+2)^2. - R. J. Mathar, Sep 11 2019

Corrected and extended by Robert G. Wilson v, Feb 12 2005

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A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

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A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.

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A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

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A110522 Riordan array (1/(1+x), x*(1-2*x)/(1+x)^2).

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Magma

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A103646 G.f.: 9*(3x+1)/(1+2x-6x^2-27x^3).

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A110522 Riordan array (1/(1+x), x(1-2x)/(1+x)^2).