A006234 -id:A006234 - cp's OEIS frontend

A081280 Binomial transform of Chebyshev coefficients A006974.

1, 10, 69, 398, 2057, 9858, 44685, 194022, 813969, 3319866, 13224789, 51635070, 198148761, 749016882, 2794021533, 10300389462, 37575535905, 135782112618, 486470994021, 1729358969454, 6104068084521, 21404982017250, 74609825192109

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 (15,-90,270,-405,243).

Cf. A007051, A006234, A081279, A081281.

[(n^4+24*n^3+164*n^2+378*n+243)*3^(n-5): n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[(1 - 2 x) (1 - x)^3 / (1 - 3 x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{15,-90,270,-405,243},{1,10,69,398,2057},30] (* Harvey P. Dale, May 05 2019 *)

a(n) = (n^4+24*n^3+164*n^2+378*n+243) * 3^(n-5).
a(n) = 15*a(n-1) -90*a(n-2) +270a*(n-3) -405*a(n-4) +243*a(n-5).
G.f.: (1-2*x)*(1-x)^3/(1-3*x)^5.

A082308 Expansion of e.g.f. (1+x)exp(4x)*cosh(x).

Original entry on oeis.org

1, 5, 25, 127, 657, 3449, 18281, 97395, 519841, 2773741, 14776377, 78538343, 416367665, 2201517153, 11610231433, 61078202971, 320570884929, 1678897264085, 8775159682649, 45780628812879, 238431945108433

Offset: 0

Paul Barry, Apr 09 2003

Binomial transform of A082307.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-94,240,-225).

Cf. A082309.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(4*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[4*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

x='x+O('x^30); Vec(serlaplace((1+x)*exp(4*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

a(n) = (A081105(n) + A006234(n))/2.
a(n) = ((n+3)*3^(n-1) + (n+5)*5^(n-1))/2.
G.f.: ((1-4*x)/(1-5*x)^2 + (1-2*x)/(1-3*x)^2)/2.
E.g.f.: (1+x)*exp(4*x)*cosh(x) = (1+x)*(exp(5*x) + exp(3*x))/2.

A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704, 99, 11, 1

Offset: 0

Paul D. Hanna, Nov 23 2003

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.

			Rows begin:
  {1, 2, 3, 4, 5, 6, 7,..},
  {1, 3, 8, 20, 48, 112, 256,..},
  {1, 4, 15, 54, 189, 648, 2187,..},
  {1, 5, 24, 112, 512, 2304, 10240,..},
  {1, 6, 35, 200, 1125, 6250, 34375,..},
  {1, 7, 48, 324, 2160, 14256, 93312,..},
  {1, 8, 63, 490, 3773, 28812, 218491,..},..

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals).

Cf. A089945, A089946.
Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.
Columns : A000012, A000027, A005563.

A089944[n_, k_] := (k + n + 1)*(n + 1)^(k - 1);
Table[A089944[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

T(n,k)=if(n<0 || k<0,0,(k+n+1)*(n+1)^(k-1))

T(n,k) = (k+n+1)*(n+1)^(k-1).

E.g.f.: (1+x)*exp(x)/(1-y*exp(x)).

A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

Original entry on oeis.org

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

Offset: 0

Mark Dols, Sep 01 2009

Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

			As triangle:
    1;
    1,   -1;
    3,   -4,    1;
    9,  -15,    7,   -1;
   27,  -54,   36,  -10,    1;
   81, -189,  162,  -66,   13,   -1;
  243, -648,  675, -360,  105,  -16,    1;

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

Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.
Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.
Cf. A133494, A136158, A164942.

A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
[A164948(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023

A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2023 *)

def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Sum_{k=0..n} T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
From Philippe Deléham, Oct 09 2011: (Start)
T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.
Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
G.f.: (1-2*x)/(1-3*x+x*y). - R. J. Mathar, Aug 12 2015
From G. C. Greubel, Dec 26 2023: (Start)
T(n, k) = (-1)^k * A136158(n, k).
T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
T(n, 0) = A133494(n).
T(n, 1) = -A006234(n+2), n >= 1.
T(n, 2) = A080420(n-2), n >= 2.
T(n, 3) = -A080421(n-3), n >= 3.
T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.
T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
T(n, n) = (-1)^n.
Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)

More terms from Philippe Deléham, Oct 09 2011

A193731 Mirror of the triangle A193730.

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 9, 30, 28, 8, 27, 108, 144, 80, 16, 81, 378, 648, 528, 208, 32, 243, 1296, 2700, 2880, 1680, 512, 64, 729, 4374, 10692, 14040, 10800, 4896, 1216, 128, 2187, 14580, 40824, 63504, 60480, 36288, 13440, 2816, 256, 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512

Offset: 0

Clark Kimberling, Aug 04 2011

A193731 is obtained by reversing the rows of the triangle A193730.

Triangle T(n,k), read by rows, given by (1,2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   1,   2;
   3,   8,   4;
   9,  30,  28,   8;
  27, 108, 144,  80,  16;
  81, 378, 648, 528, 208, 32;

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

Cf. A000012, A000079, A005053, A006234, A007483, A080420, A080421.
Cf. A080422, A080423, A084938, A130129, A133494, A153881.
Cf. A193722, A193730.

function T(n, k) // T = A193731
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023

(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1, k] + 2*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)

def T(n, k): # T = A193731
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

T(n,k) = A193730(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x)/(1-3*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = 2*A006234(n+2).
T(n, 2) = 4*A080420(n-2).
T(n, 3) = 8*A080421(n-3).
T(n, 4) = 16*A080422(n-4).
T(n, 5) = 32*A080423(n-5).
T(n, n) = A000079(n).
T(n, n-1) = A130129(n-1).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A153881(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007483(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000012(n). (End)

A079027 a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i)=6, m(i,j)=i/j.

Original entry on oeis.org

6, 35, 200, 1125, 6250, 34375, 187500, 1015625, 5468750, 29296875, 156250000, 830078125, 4394531250, 23193359375, 122070312500, 640869140625, 3356933593750, 17547607421875, 91552734375000, 476837158203125, 2479553222656250, 12874603271484375

Offset: 1

Benoit Cloitre, Feb 01 2003

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

Cf. A006234.

A079027:=n->(n + 5)*5^(n - 1); seq(A079027(n), n=1..50); # Wesley Ivan Hurt, Nov 30 2013

Table[(n + 5)*5^(n - 1), {n, 50}] (* Wesley Ivan Hurt, Nov 30 2013 *)
LinearRecurrence[{10,-25},{6,35},30] (* Harvey P. Dale, Jun 14 2022 *)

a(n) = matdet(matrix(n, n, i, j, if(i==j, 6, i/j))); \\ Michel Marcus, Nov 30 2013

a(n) = (n+5)*5^(n-1).
a(n) = 10*a(n-1)-25*a(n-2). G.f.: -x*(25*x-6) / (5*x-1)^2. - Colin Barker, Jun 18 2013
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 15625*log(5/4) - 41837/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = 34187/12 - 15625*log(6/5). (End)

More terms from Michel Marcus, Nov 30 2013

A082306 Expansion of e.g.f. (1+x)exp(2x)*cosh(x).

Original entry on oeis.org

1, 3, 9, 29, 97, 327, 1097, 3649, 12033, 39371, 127945, 413349, 1328609, 4251535, 13551753, 43046729, 136314625, 430467219, 1355971721, 4261625389, 13366006881, 41841412823, 130754415049, 407953774929, 1270932914177

Offset: 0

Paul Barry, Apr 09 2003

Binomial transform of A082305 a(n)=(A006234(n)+A000027(n))/2

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Cf. A082307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(2*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[2*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

x='x+O('x^30); Vec(serlaplace((1+x)*exp(2*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

a(n) = (n + 1 + 3^(n-1)*(n + 3))/2.
G.f.: (1/(1-x)^2 + (1-2*x)/(1-3*x)^2)/2.
E.g.f.: (1+x)*exp(2*x)*cosh(x).

A125586 a(n) = 2^(2n-1) - (n+2)*3^(n-2).

Original entry on oeis.org

1, 4, 17, 74, 323, 1400, 6005, 25478, 107015, 445556, 1841273, 7561922, 30897227, 125714672, 509767421, 2061390206, 8317305359, 33498803948, 134727010049, 541232563130, 2172291241811, 8712410196584, 34922863258757, 139921580805494, 560408087592983

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Jan 05 2007

Number of n X n nonsingular real matrices with entries {0,1} in which the top left n-1 X n-1 submatrix is the identity matrix. See A125587 for proof.

The number of singular matrices is given by A006234.

			a(2) = 4:
10 10 11 11
01 11 01 10

Index entries for linear recurrences with constant coefficients, signature (10,-33,36).

Cf. A125587, A006234.

A125586:=n->2^(2n-1)-(n+2)*3^(n-2); seq(A125586(n), n=1..30); # Wesley Ivan Hurt, Feb 26 2014

Table[2^(2n-1)-(n+2)*3^(n-2), {n, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)
LinearRecurrence[{10,-33,36},{1,4,17},50] (* Harvey P. Dale, Sep 15 2019 *)

Vec(-x*(10*x^2-6*x+1)/((3*x-1)^2*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 26 2014

G.f.: -x*(10*x^2-6*x+1) / ((3*x-1)^2*(4*x-1)). - Colin Barker, Feb 26 2014

A291004 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - 3*S)^2.

Original entry on oeis.org

6, 33, 168, 816, 3840, 17664, 79872, 356352, 1572864, 6881280, 29884416, 128974848, 553648128, 2365587456, 10066329600, 42681237504, 180388626432, 760209211392, 3195455668224, 13400297963520, 56075093016576, 234195976716288, 976366325465088, 4063794976260096

Offset: 0

Clark Kimberling, Aug 23 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A000012, A289780, A291000.

[3*(4^(n-1)*(3*n+8)): n in [0..30]]; // Vincenzo Librandi, Aug 27 2017

z = 60; s = x/(1-x); p = (1 - 3 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291004 *)
LinearRecurrence[{8, -16}, {6, 33}, 25] (* Vincenzo Librandi, Aug 27 2017 *)

[3*4^n*(3*n+8)//4 for n in range(41)] # G. C. Greubel, Jun 01 2023

G.f.: 3*(2 - 5*x)/(1 - 4*x)^2.
a(n) = 8*a(n-1) - 16*a(n-2) for n >= 3.
a(n) = 3*A006234(n+3) for n >= 0.
a(n) = 3 * 4^(n-1) * (3*n+8). - Colin Barker, Aug 23 2017

A382618 a(n) = 3^(n-2)(binomial(n,2) + 3n + 9).

Original entry on oeis.org

1, 4, 16, 63, 243, 918, 3402, 12393, 44469, 157464, 551124, 1909251, 6554439, 22320522, 75464622, 253497357, 846585513, 2812385772, 9298091736, 30606218631, 100341906651, 327757733694, 1066956026706, 3462376910193, 11203038280413, 36150980669568, 116360969030172

Offset: 0

Enrique Navarrete, Apr 01 2025

a(n) is the number of words of length n defined on 4 letters where a chosen letter (for example, the first letter of the alphabet) is used at most twice.

			a(4) = 243 since from the 256 words defined on {0, 1, 2, 3} we subtract the 4 words of type 0001, the 4 words of type 0002, the 4 words of type 0003, and 0000.

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A006234.

a[n_] := 3^(n-2)*(n^2 + 5*n + 18)/2; Array[a, 27, 0] (* Amiram Eldar, Apr 01 2025 *)

E.g.f.: (1 + x + x^2/2)*exp(3*x).

G.f.: x*(1 - 5*x + 7*x^2)/(1 - 3*x)^3. - Stefano Spezia, Apr 01 2025

cp's OEIS Frontend

A081280 Binomial transform of Chebyshev coefficients A006974.

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A082308 Expansion of e.g.f. (1+x)*exp(4*x)*cosh(x).

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A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

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A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

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A193731 Mirror of the triangle A193730.

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A079027 a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i)=6, m(i,j)=i/j.

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A082306 Expansion of e.g.f. (1+x)*exp(2*x)*cosh(x).

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A082308 Expansion of e.g.f. (1+x)exp(4x)*cosh(x).

A082306 Expansion of e.g.f. (1+x)exp(2x)*cosh(x).

A382618 a(n) = 3^(n-2)(binomial(n,2) + 3n + 9).