A057084 -id:A057084 - cp's OEIS frontend

A058404 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058405.

1, 8, 22, 56, 376, 588, 384, 4576, 17024, 19656, 2624, 48256, 313504, 848096, 801360, 17920, 468608, 4643072, 21685888, 47494272, 38797920, 122368, 4307456, 60136448, 424509952, 1590913920, 2986217856, 2181332160, 835584, 38055936

Offset: 0

Wolfdieter Lang, Dec 11 2000

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of P0(n) := A000129(n+1), n >= 0, (Pell numbers starting with P0(0)=1) with itself is Pk(n) := A054456(n+k,k) = (p(k-1,n)*(n+1)*2*P0(n+1) + q(k-1,n)*(n+2)*P0(n))/(k!*8^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058405(k,m).
a(k,0)= A057084(k), k >= 0 (conjecture).

			k=2: P2(n)=(8*n+22)*(n+1)*2*P0(n+1)+(8*n+20)*(n+2)*P0(n))/128, cf. A054457.
1; 8,22; 56,376,588; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0)

W. Lang, First 7 rows, also for A058405.

Cf. A000129, A054456, A058405, A054457, A057084, A058402-3 (rising powers).

Recursion for row polynomials defined in the comments: see A058402.

Link and cross-references added by Wolfdieter Lang, Jul 31 2002

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, -1, 0, 1, 4, 6, 0, -1, 0, 1, 5, 12, 9, -4, 0, 0, 1, 6, 20, 32, 9, -8, 1, 0, 1, 7, 30, 75, 80, 0, -8, 1, 0, 1, 8, 42, 144, 275, 192, -27, 0, 0, 0, 1, 9, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1,  1,  1, 1,   1,    1, ...
  0,  1,  2, 3,   4,    5, ...
  0,  0,  2, 6,  12,   20, ...
  0, -1,  0, 9,  32,   75, ...
  0, -1, -4, 9,  80,  275, ...
  0,  0, -8, 0, 192, 1000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Rows 0..1 give A000012, A001477.
Main diagonal gives (-1) * A109519(n+1).
Cf. A342120, A342133, A342134.

T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A164608 Expansion of (1+4x)/(1-8x+8*x^2).

Original entry on oeis.org

1, 12, 88, 608, 4160, 28416, 194048, 1325056, 9048064, 61784064, 421888000, 2880831488, 19671547904, 134325731328, 917233467392, 6263261888512, 42768227368960, 292039723843584, 1994171971796992, 13617057983627264

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Binomial transform of A054490. Fourth binomial transform of A164683. Inverse binomial transform of A164609.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..149 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-8).

Cf. A054490, A164683, A164609.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(4+2*r)^n+(2-4*r)*(4-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009

LinearRecurrence[{8,-8}, {1,12,88}, 50] (* G. C. Greubel, Aug 10 2017 *)

x='x+O('x^50); Vec((1+4*x)/(1-8*x+8*x^2)) \\ G. C. Greubel, Aug 10 2017

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 12.
a(n) = A057084(n) + 4*A057084(n-1).
a(n) = ((2+4*sqrt(2))*(4+2*sqrt(2))^n + (2-4*sqrt(2))*(4-2*sqrt(2))^n)/4.
E.g.f.: exp(4*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

A365823 Decimal expansion of 2*(2 + sqrt(2)).

Original entry on oeis.org

6, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7

Offset: 1

Wolfdieter Lang, Nov 13 2023

The greater one of the solutions to x^2 - 8 * x + 8 = 0. The other solution is A157259 - 3 = 1.17157... . - Michal Paulovic, Nov 14 2023

			6.8284271247461900976033774484193961571393437507538961...

Cf. A000129, A002193, A014176, A049310, A057084, A157259.
Cf. A156035, A163960.
Essentially the same as A157258, A090488, A086178 and A010466.

evalf(4+sqrt(8), 130);  # Alois P. Heinz, Nov 13 2023

First[RealDigits[2*(2 + Sqrt[2]), 10, 99]] (* Stefano Spezia, Nov 11 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^n*(4+quadgen(32)))) \\ Michal Paulovic, Nov 14 2023

Equals 2*sqrt(2)*(1 + sqrt(2)) = 2*(2 + sqrt(2)). This is an integer in the quadratic number field Q(sqrt(2)).
Equals lim_{n->oo} A057084(n + 1)/A057084(n).
Equals continued fraction with periodic term [[6], [1, 4]]. - Peter Luschny, Nov 13 2023
Equals -3+A157258 = 1+A156035 = 2+A090488 = 3+A086178 = 4+A010466 = 6+A163960. - Alois P. Heinz, Nov 15 2023

A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 2*x^2.

Original entry on oeis.org

1, 1, 3, 2, 3, 7, 3, 9, 5, 15, 5, 15, 26, 3, 31, 8, 30, 43, 63, -15, 63, 13, 54, 104, 87, 144, -81, 127, 21, 99, 203, 273, 115, 333, -275, 255, 34, 177, 416, 549, 609, -9, 806, -789, 511, 55, 315, 811, 1263, 1146, 1260, -725, 2043, -2071, 1023, 89, 555, 1573

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    3
   2    3     7
   3    9     5    15
   5   15    26     3    31
   8   30    43    63   -15    63
  13   54   104    87   144   -81    127
  21   99   203   273   115   333   -275   255
Row 4 represents the polynomial p(4,x) = 3 + 9*x + 5*x^2 + 15*x^3, so (T(4,k)) = (3,9,5,15), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Cf. A000045 (column 1); A000225, (p(n,n-1)); A001787 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004254, (p(n,-2)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368156.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - 3*x - 2*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A123357 Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

Original entry on oeis.org

1, 6, 40, 272, 1856, 12672, 86528, 590848, 4034560, 27549696, 188121088, 1284571136, 8771600384, 59896233984, 408997068800, 2792806678528, 19070476877824, 130221361594368, 889207077732352, 6071885729103872, 41461429210972160, 283116347854946304, 1933239349151793152

Offset: 0

N. J. A. Sloane, Oct 10 2006

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 210, formula page 204).
Index entries for linear recurrences with constant coefficients, signature (8,-8).

A123357 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,6]) ;
    else
        8*(procname(n-1)-procname(n-2)) ;
    end if
end proc:
seq( A123357(n),n=0..30) ; # R. J. Mathar, Jul 26 2019

LinearRecurrence[{8, -8}, {1, 6}, 30] (* Jean-François Alcover, Apr 03 2020 *)

G.f.: -(2*x-1) / (8*x^2-8*x+1). - Colin Barker, Aug 29 2013

a(n) = A057084(n)-2*A057084(n-1). - R. J. Mathar, Jul 26 2019

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.

Original entry on oeis.org

1, 9, 73, 577, 4529, 35481, 277817, 2174993, 17027041, 133295529, 1043495593, 8168931937, 63949894289, 500627099961, 3919122796697, 30680567267633, 240180585132481, 1880236207775049, 14719292130498313, 115228905772807297, 902061091509601649

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

nonn

Binomial transform of A057084. Second binomial transform of A002315. Third binomial transform of A108051 without initial 0. Fourth binomial transform of A096980. Fifth binomial transform of A094015.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17).

Cf. A057084, A002315, A108051, A096980, A094015.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+3*r)*(5+2*r)^n+(3-3*r)*(5-2*r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

LinearRecurrence[{10,-17},{1,9},30] (* Harvey P. Dale, Sep 11 2016 *)

x='x+O('x^50); Vec((1-x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1-x)/(1-10*x+17*x^2).
E.g.f.: (1/3)*exp(5*x)*(3*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

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

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A114479 Kekulé numbers for certain benzenoids.

Original entry on oeis.org

3, 20, 136, 928, 6336, 43264, 295424, 2017280, 13774848, 94060544, 642285568, 4385800192, 29948116992, 204498534400, 1396403339264, 9535238438912, 65110680797184, 444603538866176, 3035942864551936, 20730714605486080

Offset: 1

Emeric Deutsch, Nov 30 2005

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 205).

Index entries for linear recurrences with constant coefficients, signature (8,-8).

a:=((4+sqrt(8))^(n+1)+(4-sqrt(8))^(n+1))/16: seq(expand(a(n)),n=1..23);

a(n) = ((4+sqrt(8))^(n+1) + (4-sqrt(8))^(n+1))/16.
a(n) = 8*a(n-1) - 8*a(n-2). - Colin Barker, Aug 30 2013
G.f.: -x*(4*x-3) / (8*x^2 - 8*x + 1). - Colin Barker, Aug 30 2013
a(n)= 3*A057084(n-1) - 4*A057084(n-2). - R. J. Mathar, Aug 30 2013
a(n) = A007052(n+1)*2^(n-1). - R. J. Mathar, Jul 24 2022

A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 12, 28, 20, 4, 29, 88, 94, 40, 5, 70, 262, 372, 244, 70, 6, 169, 752, 1333, 1184, 539, 112, 7, 408, 2104, 4472, 5016, 3144, 1064, 168, 8, 985, 5776, 14316, 19408, 15526, 7344, 1932, 240, 9

Offset: 0

Philippe Deléham, Dec 07 2011

Diagonal sums: A201967(n), row sums: A000302(n) (powers of 4).

			Triangle begins:
    1;
    2,   2;
    5,   8,    3;
   12,  28,   20,    4;
   29,  88,   94,   40,   5;
   70, 262,  372,  244,  70,   6;
  169, 752, 1333, 1184, 539, 112, 7;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000027, A000129, A006645, A007290.

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k<0 or  k>n  then 0
    else 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

With[{m = 8}, CoefficientList[CoefficientList[Series[1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k) = if(nMichel Marcus, Feb 17 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0 and n==0): return 1
    else: return 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020

G.f.: 1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A000302(n), A138395(n), A057084(n) for x = -1, 0, 1, 2, 3, respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000027(n), A000302(n), A090018(n), A057091(n) for x = 0, 1, 2, 3, respectively.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

a(40) corrected by Georg Fischer, Feb 17 2020

cp's OEIS Frontend

A058404 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058405.

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

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A164608 Expansion of (1+4*x)/(1-8*x+8*x^2).

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A365823 Decimal expansion of 2*(2 + sqrt(2)).

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A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - 2*x^2.

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A123357 Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

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A164588 a(n) = ((3 + sqrt(18))*(5 + sqrt(8))^n + (3 - sqrt(18))*(5 - sqrt(8))^n)/6.

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

A164608 Expansion of (1+4x)/(1-8x+8*x^2).

A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 2*x^2.

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.