A052541 -id:A052541 - cp's OEIS frontend

A317497 Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

1, 3, 9, 27, 1, 81, 6, 243, 27, 729, 108, 1, 2187, 405, 9, 6561, 1458, 54, 19683, 5103, 270, 1, 59049, 17496, 1215, 12, 177147, 59049, 5103, 90, 531441, 196830, 20412, 540, 1, 1594323, 649539, 78732, 2835, 15, 4782969, 2125764, 295245, 13608, 135, 14348907, 6908733, 1082565, 61236, 945, 1

Offset: 0

Zagros Lalo, Jul 31 2018

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and  along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
The coefficients in the expansion of 1/(1-3x-x^3) are given by the sequence generated by the row sums.
The row sums give A052541.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.1038034027355..., when n approaches infinity.

			Triangle begins:
         1;
         3;
         9;
        27,        1;
        81,        6;
       243,       27;
       729,      108,       1;
      2187,      405,       9;
      6561,     1458,      54;
     19683,     5103,     270,      1;
     59049,    17496,    1215,     12;
    177147,    59049,    5103,     90;
    531441,   196830,   20412,    540,    1;
   1594323,   649539,   78732,   2835,   15;
   4782969,  2125764,  295245,  13608,  135;
  14348907,  6908733, 1082565,  61236,  945,  1;
  43046721, 22320522, 3897234, 262440, 5670, 18;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 364-366

G. C. Greubel, Rows n = 0..120 of the irregular triangle, flattened
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n

Row sums give A052541.
Cf. A013610, A027465, A317496, A304236, A304249.
Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3), A036217 (column 4).
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), this sequence (q=3), A318773 (q=4).

Flat(List([0..16],n->List([0..Int(n/3)],k->3^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Aug 01 2018

[3^(n-3*k)*Binomial(n-2*k,k): k in [0..Floor(n/3)], n in [0..24]]; // G. C. Greubel, May 12 2021

T[n_, k_]:= T[n, k] = 3^(n-3k)(n-2k)!/((n-3k)! k!); Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]//Flatten
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3 T[n-1, k] + T[n-3, k-1]]; Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}]//Flatten

flatten([[3^(n-3*k)*binomial(n-2*k,k) for k in (0..n//3)] for n in (0..24)]) # G. C. Greubel, May 12 2021

T(n,k) = 3^(n-3*k) * (n-2*k)!/(k! * (n-3*k)!) where n is a nonnegative integer and k = 0..floor(n/3).

A117716 Triangle T(n,k) read by rows: the coefficient [x^n] of x^2/(1-(k+1)*x-x^3) in row n, columns 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 1, 4, 9, 16, 25, 2, 9, 28, 65, 126, 217, 3, 20, 87, 264, 635, 1308, 2415, 4, 44, 270, 1072, 3200, 7884, 16954, 32960, 6, 97, 838, 4353, 16126, 47521, 119022, 264193, 534358, 9, 214, 2601, 17676, 81265, 286434, 835569, 2117656, 4815801, 10050030

Offset: 0

Roger L. Bagula, Apr 13 2006, corrected Apr 15 2006

			Triangle begins as:
  0;
  0,  0;
  1,  1,   1;
  1,  2,   3,    4;
  1,  4,   9,   16,   25;
  2,  9,  28,   65,  126,  217;
  3, 20,  87,  264,  635, 1308,  2415;
  4, 44, 270, 1072, 3200, 7884, 16954, 32960;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000930 (column 0), A008998 (column 1), A052541 (column 2), A052927 (column 3), A001093 (row 5), A185065 (row 6), A117715, A117724.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117716:= func< n,k | Coefficient(R!( x^2/(1-(k+1)*x-x^3) ), n) >;
[[A117716(n,k): k in [0..n]]: n in [0..m]]; // G. C. Greubel, Jul 23 2023

A117716 := proc(n,m)
        x^2/(1-(m+1)*x-x^3) ;
        if n < 0 then
                0;
        else
                coeftayl(%,x=0,n) ;
        end if;
end proc: # R. J. Mathar, May 14 2013

T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x-x^3), {x,0,n+ 2}], x, n];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A117716(n,k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x-x^3) ).list()[n]
flatten([[A117716(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

Edited by G. C. Greubel, Jul 23 2023

A098590 a(n) = 4^n for n = 0..3; for n > 3, a(n) = 4*a(n-1) + a(n-4).

Original entry on oeis.org

1, 4, 16, 64, 257, 1032, 4144, 16640, 66817, 268300, 1077344, 4326016, 17370881, 69751824, 280084640, 1124664576, 4516029185, 18133868564, 72815558896, 292386900160, 1174063629825, 4714388387864, 18930369110352, 76013863341568

Offset: 0

Paul Barry, Sep 16 2004

a(n) equals the number of n-length words on {0,1,2,3,4} such that 0 appears only in a run which length is a multiple of 4. - Milan Janjic, Feb 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,0,1).

Cf. A052541.

I:=[0,1,4,16]; [n le 4 select I[n] else 4*Self(n-1) +Self(n-4): n in [1..30]]; // G. C. Greubel, Feb 03 2018

K:=1/(1+4*z-z^4): Kser:=series(K, z=0, 30): seq(abs(coeff(Kser, z, n)), n= 0..23); # Zerinvary Lajos, Nov 08 2007

CoefficientList[Series[1/(1 - 4*x - x^4), {x, 0, 25}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{4,0,0,1},{0,1,4,16},30] (* Harvey P. Dale, Jul 22 2014 *)

x='x+O('x^30); Vec(1/(1-4*x-x^4)) \\ G. C. Greubel, Feb 03 2018

G.f.: 1/(1-4*x-x^4).

a(n) = Sum_{k=0..floor(n/3)} binomial(n-3*k, k) * 4^(n-4*k).

A052668 Expansion of e.g.f. 1/(1 - 3*x - x^3).

Original entry on oeis.org

1, 3, 18, 168, 2088, 32400, 603360, 13109040, 325503360, 9092684160, 282219033600, 9635476435200, 358879494758400, 14480588157235200, 629228583138355200, 29295027261916416000, 1454816084780298240000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..375
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 616

Cf. A052541.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( 1/(1-3*x-x^3) ))); // G. C. Greubel, Sep 03 2022

spec := [S,{S=Sequence(Union(Z,Z,Z,Prod(Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

a[n_]:= a[n]= If[n<3, 3^n*n!, 3*n*a[n-1] + n*(n-1)*(n-2)*a[n-3]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Sep 03 2022 *)

def A052668_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(1-3*x-x^3) ).egf_to_ogf().list()
A052668_list(40) # G. C. Greubel, Sep 03 2022

E.g.f.: 1/(1-3*x-x^3).
a(n) = 3*n*a(n-1) + n*(n-1)*(n-2)*a(n-3), a(0)=1, a(1)=3, a(2)=18.
a(n) = (n!/15) * Sum_{alpha=RootOf(-1+3*_Z+_Z^3)} (4 + alpha + 2*alpha^2) * alpha^(-1-n).
a(n) = n!*A052541(n). - R. J. Mathar, Nov 27 2011

A098589 a(n) = 3a(n-1) + 2a(n-3), with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 9, 29, 93, 297, 949, 3033, 9693, 30977, 98997, 316377, 1011085, 3231249, 10326501, 33001673, 105467517, 337055553, 1077170005, 3442445049, 11001446253, 35158678769, 112360926405, 359085671721, 1147574372701

Offset: 0

Paul Barry, Sep 16 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,2).

Cf. A052541.

I:=[1,3,9]; [n le 3 select I[n] else 3*Self(n-1) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 03 2018

CoefficientList[Series[1/(1-3*x-2*x^3), {x,0,50}], x] (* or *) LinearRecurrence[{3,0,2},{1,3,9}, 50] (* G. C. Greubel, Feb 03 2018 *)

x='x+O('x^30); Vec(1/(1-3*x-2*x^3)) \\ G. C. Greubel, Feb 03 2018

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

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n} Sum_{j=0..i} C(n-2*k,i)*C(i,j) *C(j, k)*2^k.

A183188 a(n) = 3*a(n-1) + a(n-3) with a(0)=1, a(1)=2, a(2)=6.

Original entry on oeis.org

1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, 4892277, 15184666, 47130218, 146282931, 454033459, 1409230595, 4373974716, 13575957607, 42137103416, 130785284964, 405931812499, 1259932540913, 3910582907703, 12137680535608

Offset: 0

Philippe Deléham, Dec 14 2011

Index entries for linear recurrences with constant coefficients, signature (3,0,1).

Cf. A052541, A193723.

LinearRecurrence[{3,0,1},{1,2,6},30] (* Harvey P. Dale, Nov 02 2024 *)

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

a(n) = A052541(n) - A051541(n-1). - R. J. Mathar, Dec 15 2011

Corrected by R. J. Mathar, Dec 15 2011

cp's OEIS Frontend

A317497 Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

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A117716 Triangle T(n,k) read by rows: the coefficient [x^n] of x^2/(1-(k+1)*x-x^3) in row n, columns 0 <= k <= n.

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A098590 a(n) = 4^n for n = 0..3; for n > 3, a(n) = 4*a(n-1) + a(n-4).

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A052668 Expansion of e.g.f. 1/(1 - 3*x - x^3).

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A098589 a(n) = 3*a(n-1) + 2*a(n-3), with a(0)=1, a(1)=3.

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A183188 a(n) = 3*a(n-1) + a(n-3) with a(0)=1, a(1)=2, a(2)=6.

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A098589 a(n) = 3a(n-1) + 2a(n-3), with a(0)=1, a(1)=3.