A109466 -id:A109466 - cp's OEIS frontend

A100335 An inverse Catalan transform of J(2n).

0, 1, 4, 11, 27, 64, 149, 341, 768, 1707, 3755, 8192, 17749, 38229, 81920, 174763, 371371, 786432, 1660245, 3495253, 7340032, 15379115, 32156331, 67108864, 139810133, 290805077, 603979776, 1252698795, 2594876075, 5368709120

Offset: 0

Paul Barry, Nov 17 2004

The g.f. is obtained from that of A002450 through the mapping g(x) -> g(x*(1-x)). A002450 may be retrieved through the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4).

Cf. A000108, A001045, A002450, A010892.

Cf. A042965, A100334, A109466, A133110.

I:=[0,1,4,11]; [n le 4 select I[n] else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4): n in [1..41]]; // G. C. Greubel, Jan 24 2023

LinearRecurrence[{5,-9,8,-4}, {0,1,4,11}, 41] (* G. C. Greubel, Jan 24 2023 *)

def A100335(n): return (1/3)*((n+1)*2^n - chebyshev_U(n,1/2))
[A100335(n) for n in range(41)] # G. C. Greubel, Jan 24 2023

G.f.: x*(1-x)/(1 - 5*x + 9*x^2 - 8*x^3 + 4*x^4).
a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(4^(n-k) - 1)/3.
a(n) = (1/3)*((n+1)*2^n - A010892(n)). - Ralf Stephan, May 15 2007
Binomial transform of A042965: (1, 3, 4, 5, 7, 8, 9, 11, 12, 13, ...), also row sums of triangle A133110. - Gary W. Adamson, Sep 12 2007
a(n) = Sum_{k=0..n} A109466(n,k)*A002450(k). - Philippe Deléham, Oct 30 2008

A110517 Riordan array (1,x(1-3x)).

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 0, -6, 1, 0, 0, 9, -9, 1, 0, 0, 0, 27, -12, 1, 0, 0, 0, -27, 54, -15, 1, 0, 0, 0, 0, -108, 90, -18, 1, 0, 0, 0, 0, 81, -270, 135, -21, 1, 0, 0, 0, 0, 0, 405, -540, 189, -24, 1, 0, 0, 0, 0, 0, -243, 1215, -945, 252, -27, 1, 0, 0, 0, 0, 0, 0, -1458, 2835, -1512, 324, -30, 1, 0, 0, 0, 0, 0, 0, 729, -5103, 5670, -2268, 405

Offset: 0

Paul Barry, Jul 24 2005

Inverse is Riordan array (1,xc(3x)) [A110518]. Row sums are A106852. Diagonal sums are A106855.

Modulo 2, this sequence becomes A106344. - Philippe Deléham, Dec 19 2008

			Rows begin
1;
0, 1;
0, -3, 1;
0, 0, -6, 1;
0, 0, 9, -9, 1;
0, 0, 0, 27, -12, 1;
0, 0, 0, -27, 54, -15, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, k_] := (-3)^(n - k)*Binomial[k, n - k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

for(n=0,20, for(k=0,n, print1((-3)^(n-k)*binomial(k, n-k), ", "))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(n, k) = (-3)^(n-k)*binomial(k, n-k).

T(n,k) = A109466(n,k)*3^(n-k). - Philippe Deléham, Oct 26 2008

A146078 Expansion of 1/(1-x(1-9x)).

Original entry on oeis.org

1, 1, -8, -17, 55, 208, -287, -2159, 424, 19855, 16039, -162656, -307007, 1156897, 3919960, -6492113, -41771753, 16657264, 392603041, 242687665, -3290739704, -5474928689, 24141728647, 73416086848, -143859470975, -804604252607

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1, x(1-9x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-9).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1, -9}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ G. C. Greubel, Jan 19 2018

[lucas_number1(n,1,9) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 9*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+9*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)
a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022

A146080 Expansion of 1/(1-x(1-10x)).

Original entry on oeis.org

1, 1, -9, -19, 71, 261, -449, -3059, 1431, 32021, 17711, -302499, -479609, 2545381, 7341471, -18112339, -91527049, 89596341, 1004866831, 108903421, -9939764889, -11028799099, 88368849791, 198656840781, -685031657129, -2671600064939

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-10x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-10).

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

CoefficientList[Series[1/(1-x(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{1,-10},{1,1},30] (* Harvey P. Dale, Dec 16 2012 *)

Vec(1/(1-x*(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

[lucas_number1(n,1,10) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 10*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*10^(n-k).
E.g.f.: exp(x/2)*(cos(sqrt(39)*x/2) + (1/sqrt(39))*sin(sqrt(39)*x/2)). - G. C. Greubel, Jan 30 2016

A106233 An inverse Catalan transform of A003462.

Original entry on oeis.org

0, 1, 3, 5, 5, 0, -14, -41, -81, -121, -121, 0, 364, 1093, 2187, 3281, 3281, 0, -9842, -29525, -59049, -88573, -88573, 0, 265720, 797161, 1594323, 2391485, 2391485, 0, -7174454, -21523361, -43046721, -64570081, -64570081, 0, 193710244, 581130733, 1162261467

Offset: 0

Paul Barry, Apr 26 2005

The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).

The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012

			From _Paul Curtz_, Nov 09 2012: (Start)
The sequence and its higher-order differences (periodic after 6 rows):
   0,  1,  3,  5,  5,   0, -14, ...
   1,  2,  2,  0, -5, -14, -27, ...
   1,  0, -2, -5, -9, -13, -13, ...
  -1, -2, -3, -4, -4,   0,  13, ...   = -A134581(n+1)
  -1, -1, -1,  0,  4,  13,  27, ...
   0,  0,  1,  4,  9,  14,  14, ...   = A140343(n+2)
   0,  1,  3,  5,  5,   0, -14, ...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Cf. A103368.

I:=[0,1,3,5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2018

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)

G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) = Sum_{k=0..n} A109466(n,k)*A003462(k). - Philippe Deléham, Oct 30 2008
a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]6 ]. - _Ralf Stephan, Nov 15 2010
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - Paul Curtz, Nov 09 2012
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012

A146083 Expansion of 1/(1 - x(1 - 11x)).

Original entry on oeis.org

1, 1, -10, -21, 89, 320, -659, -4179, 3070, 49039, 15269, -524160, -692119, 5073641, 12686950, -43123101, -182679551, 291674560, 2301149621, -907270539, -26219916370, -16239940441, 272179139629, 450818484480, -2543152051439

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-11x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-11).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080.

[n le 2 select 1 else Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016

LinearRecurrence[{1,-11},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec(1/(1-x*(1-11*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,11) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-11*a(n-2) ; a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*11^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+11*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(43)*x/2) + (1/sqrt(43))*sin(sqrt(43)*x/2)).
(End)

A146541 Binomial transform of A010688.

Original entry on oeis.org

1, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Philippe Deléham, Oct 31 2008

Hankel transform is := 1,-48,0,0,0,0,0,0,0,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000007, A011782, A000079, A003945, A046055, A146523, A082505, A135092.

Essentially the same as A132479, A077552, A020707.

Join[{1},2^Range[3,40]] (* Harvey P. Dale, Feb 28 2016 *)

Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(0)=1, a(n) = 2^(n+2) for n>0.
a(n) = Sum_{k, 0..n} A109466(n,k)*A146534(k).
a(n) = A132479(n), n>1. - R. J. Mathar, Nov 02 2008
G.f.: (1+6*x) / (1-2*x). - Colin Barker, Mar 17 2016

Corrected and extended by Harvey P. Dale, Feb 28 2016

A099322 An inverse Catalan transform of J(3n)/J(3).

Original entry on oeis.org

0, 1, 6, 43, 291, 1992, 13595, 92845, 633966, 4329023, 29560367, 201850896, 1378323999, 9411785201, 64267689006, 438847231427, 2996636337771, 20462312853336, 139725412120339, 954104794142789, 6515035056168654

Offset: 0

Paul Barry, Nov 17 2004

The g.f. is obtained from that of A015565 through the mapping g(x)->g(x(1-x)). A015565 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108.

Index entries for linear recurrences with constant coefficients, signature (7,1,-16,8).

Cf. A001045.

LinearRecurrence[{7,1,-16,8},{0,1,6,43},30] (* Harvey P. Dale, Jul 19 2016 *)

G.f.: x(1-x)/(1-7x-x^2+16x^3-8x^4);
a(n) = 7a(n-1) + a(n-2) - 16a(n-3) + 8a(n-4);
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*J(3n-3k)/J(3).
a(n) = Sum_{k=0..n} A109466(n,k)*A015565(k). - Philippe Deléham, Oct 30 2008

A146084 Expansion of 1/(1-x(1-12x)).

Original entry on oeis.org

1, 1, -11, -23, 109, 385, -923, -5543, 5533, 72049, 5653, -858935, -926771, 9380449, 20501701, -92063687, -338084099, 766680145, 4823689333, -4376472407, -62260744403, -9743075519, 737385857317, 854302763545, -7994327524259

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-12x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -12).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080, A146083

LinearRecurrence[{1, -12}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

Vec(1/(1-x*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,12) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-12*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*12^(n-k).

A164925 Array, binomial(j-i,j), read by rising antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, 0, 0, 1, 1, -3, 1, 0, 0, 1, 1, -4, 3, 0, 0, 0, 1, 1, -5, 6, -1, 0, 0, 0, 1, 1, -6, 10, -4, 0, 0, 0, 0, 1, 1, -7, 15, -10, 1, 0, 0, 0, 0, 1, 1, -8, 21, -20, 5, 0, 0, 0, 0, 0, 1, 1, -9, 28, -35, 15, -1, 0, 0, 0, 0, 0, 1, 1, -10, 36, -56, 35, -6, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Mark Dols, Aug 31 2009

Inverse of A052509, or A004070???

			Array, A(n, k), begins as:
  1,  1,  1,   1,  1,   1,  1,  1,  1, ...
  1,  0,  0,   0,  0,   0,  0,  0,  0, ...
  1, -1,  0,   0,  0,   0,  0,  0,  0, ...
  1, -2,  1,   0,  0,   0,  0,  0,  0, ...
  1, -3,  3,  -1,  0,   0,  0,  0,  0, ...
  1, -4,  6,  -4,  1,   0,  0,  0,  0, ...
  1, -5, 10, -10,  5,  -1,  0,  0,  0, ...
  1, -6, 15, -20, 15,  -6,  1,  0,  0, ...
  1, -7, 21, -35, 35, -21,  7, -1,  0, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  0,  1;
  1, -2,  0,  0,  1;
  1, -3,  1,  0,  0,  1;
  1, -4,  3,  0,  0,  0,  1;
  1, -5,  6, -1,  0,  0,  0,  1;
  1, -6, 10, -4,  0,  0,  0,  0,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A004070, A008346, A010892, A052509, A052992.

Cf. A109466, A130595, A164899, A164915, A164965.

A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >;
[A164925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2023

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

{A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* Michael Somos, Jan 25 2012 */

def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 10 2023

Sum_{k=0..n} T(n, k) = A164965(n). - Mark Dols, Sep 02 2009
From G. C. Greubel, Feb 10 2023: (Start)
A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)

Edited by Michael Somos, Jan 26 2012

Offset changed by G. C. Greubel, Feb 10 2023

cp's OEIS Frontend

A100335 An inverse Catalan transform of J(2n).

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A110517 Riordan array (1,x(1-3x)).

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A146078 Expansion of 1/(1-x*(1-9*x)).

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A146080 Expansion of 1/(1-x*(1-10*x)).

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A106233 An inverse Catalan transform of A003462.

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A146083 Expansion of 1/(1 - x*(1 - 11*x)).

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A146541 Binomial transform of A010688.

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A146078 Expansion of 1/(1-x(1-9x)).

A146080 Expansion of 1/(1-x(1-10x)).

A146083 Expansion of 1/(1 - x(1 - 11x)).