A128504 -id:A128504 - cp's OEIS frontend

A101950 Product of A049310 and A007318 as lower triangular matrices.

1, 1, 1, 0, 2, 1, -1, 1, 3, 1, -1, -2, 3, 4, 1, 0, -4, -2, 6, 5, 1, 1, -2, -9, 0, 10, 6, 1, 1, 3, -9, -15, 5, 15, 7, 1, 0, 6, 3, -24, -20, 14, 21, 8, 1, -1, 3, 18, -6, -49, -21, 28, 28, 9, 1, -1, -4, 18, 36, -35, -84, -14, 48, 36, 10, 1, 0, -8, -4, 60, 50, -98, -126, 6, 75, 45, 11, 1, 1, -4, -30, 20, 145, 36, -210

Offset: 0

Paul Barry, Dec 22 2004

A Chebyshev and Pascal product.
Row sums are n+1, diagonal sums the constant sequence 1 resp. A023434(n+1). Riordan array (1/(1-x+x^2),x/(1-x+x^2)).
Apart from signs, identical with A104562.
Subtriangle of the triangle given by [0,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010
The Fi1 and Fi2 sums lead to A004525 and the Gi1 sums lead to A077889, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 06 2011
Also the convolution triangle of the inverse of 6th cyclotomic polynomial A010892. - Peter Luschny, Oct 08 2022

			Triangle begins:
   1,
   1, 1,
   0, 2, 1,
  -1, 1, 3, 1,
  -1,-2, 3, 4, 1,
  ...
Triangle [0,1,-1,1,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins : 1 ; 0,1 ; 0,1,1 ; 0,0,2,1 ; 0,-1,1,3,1 ; 0,-1,-2,3,4,1 ; ... - _Philippe Deléham_, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1325
Jerry Ray Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].
Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, Root geometry of polynomial sequences. II: Type (1,0), J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).

Cf. A049310, A007318, A104562, A010892, A084938.

A101950 := proc(n,k) local j,k1: add((-1)^((n-j)/2)*binomial((n+j)/2,j)*(1+(-1)^(n+j))* binomial(j,k)/2, j=0..n) end: seq(seq(A101950(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 06 2011
# Uses function PMatrix from A357368. Adds a row on top and a column to the left.
PMatrix(10, n -> [0, 1, 1, 0, -1,-1][irem(n, 6) + 1]); # Peter Luschny, Oct 08 2022

T[0, 0] = 1; T[n_, k_] /; k>n || k<0 = 0; T[n_, k_] := T[n, k] = T[n-1, k-1]+T[n-1, k]-T[n-2, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

T(n, k) = Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*(1+(-1)^(n+j))*C(j,k)/2.
T(0,0) = 1, T(n,k) = 0,if k>n or if k<0, T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010
p(n,x) = (x+1)*p(n-1,x)-p(n-2,x) with p(0,x) = 1 and p(1,x) = x+1 [Dias].
G.f.: 1/(1-x-x^2-y*x). - Philippe Deléham, Feb 10 2012
T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - Philippe Deléham, Mar 07 2014
T(n,k) = C(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4) for n>=1. - Peter Luschny, Apr 25 2016

Typo in formula corrected and information added by Johannes W. Meijer, Aug 06 2011

A128503 Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.

Original entry on oeis.org

1, 3, 6, -3, 10, -12, 15, -30, 6, 21, -60, 30, 28, -105, 90, -10, 36, -168, 210, -60, 45, -252, 420, -210, 15, 55, -360, 756, -560, 105, 66, -495, 1260, -1260, 420, -21, 78, -660, 1980, -2520, 1260, -168, 91, -858, 2970, -4620, 3150, -756, 28, 105, -1092, 4290, -7920, 6930, -2520, 252, 120, -1365

Offset: 0

Wolfdieter Lang Apr 04 2007

S2(n,x):=sum(S(n-k,x)*S1(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)) with the first convolution S1(n,x) given by array A128502.

Row polynomials P2(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).

			n=4: [15,-30,6] stands for the polynomial S2(4,x) = 15*x^4-30*x^2+6 = 2*(S(4,x)*S1(0,x)+S(3,x)*S1(1,x))+S(2,x)*S1(2,x).
n=4:[15,-30,6] stands also for the row polynomial P2(4,x) = 15-30*x+6*x^2.
[1];[3];[6,-3];[10,-12];[15,-30,6];[21,-60,30];[28,-105,90,-10];...

W. Lang, First 13 rows and more.

Row sums (signed array) give A128504. Unsigned row sums are A001628.

Cf. A128502 (k=1 convolution). A128505 (k=3 convolution).

a(n,m)= binomial(n-m+2,2)*binomial(n-m,m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)= binomial(m+2,2)*binomial(n-m+2,m+2)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S2(n,x): 1/(1-x*z+z^2)^3.
G.f. for P2(n,x): 1/(1-z+x*z^2)^3

A147621 The 3rd Witt transform of A000292.

Original entry on oeis.org

0, 0, 0, 0, 4, 26, 120, 455, 1456, 4122, 10608, 25194, 55980, 117572, 235144, 450681, 832048, 1485800, 2575368, 4345965, 7158060, 11532402, 18209100, 28224105, 43008120, 64512240, 95365920, 139075245, 200268432, 284997384, 401107356

Offset: 0

R. J. Mathar, Nov 08 2008

The 2nd Witt transform is essentially in A032094.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (8,-28,60,-102,168,-258,336,-393,452,-484,452,-393, 336,-258,168,-102,60,-28,8,-1).

Cf. A000292, A032094, A049347, A099254, A128504.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[x^4(2*x^2 - x + 2)(2*x^4 - 2*x^3 + 9*x^2 - 2*x+2)/((1-x)^12 * (1 + x + x^2)^4), {x, 0, 40}],  x] (* Vincenzo Librandi  Dec 13 2012 *)

def A147621_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) ).list()
A147621_list(40) # G. C. Greubel, Oct 24 2022

G.f.: x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4).

a(n) = (1/729)*(b(n) + c(n)), where b(n) = n*(n+3)*(n+6)*(3*n^8 +72*n^7 +618*n^6 + 2052*n^5 +207*n^4 -11772*n^3 -14268*n^2 +9648*n -232960)/492800 and c(n) = 9*A049347(n) +5*A049347(n-1) +9*(-1)^n*(A099254(n) -A099254(n-1)) -18(-1)^n*A128504(n) +27*(-1)^n*Sum_{k=0..n} A099254(n-k)*A099254(k-1). - G. C. Greubel, Oct 24 2022

A147618 The 3rd Witt transform of A000217.

Original entry on oeis.org

0, 0, 0, 0, 3, 15, 54, 165, 429, 999, 2145, 4290, 8100, 14586, 25194, 41985, 67830, 106590, 163431, 245157, 360525, 520749, 740025, 1036035, 1430703, 1950975, 2629575, 3506085, 4628052, 6052068, 7845255, 10086780, 12869340, 16301142

Offset: 0

R. J. Mathar, Nov 08 2008

The 2nd Witt transform of A000217 is essentially in A032092.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Index entries for linear recurrences with constant coefficients, signature (6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1).

Cf. A000217, A032092, A049347, A099254, A128504.

R:=PowerSeriesRing(Integers(), 30); [0,0,0,0] cat Coefficients(R!( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3), {x,0,40}], x] (* Vincenzo Librandi, Dec 13 2012 *)

def A147618_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) ).list()
A147618_list(30) # G. C. Greubel, Oct 24 2022

G.f.: 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3).

a(n) = (1/81)*(n*(n+3)*(3*n^6 +27*n^5 +45*n^4 -135*n^3 -288*n^2 +108*n -2000)/4480 +2*A049347(n) +A049347(n-1) +(-1)^n*(A099254(n) -2*A099254(n- 1)) -3*(-1)^n*(A128504(n) -2*A128504(n-1))). - G. C. Greubel, Oct 24 2022

A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

Original entry on oeis.org

1, -9, 26, -25, -36, 133, -132, -81, 375, -374, -144, 806, -805, -225, 1480, -1479, -324, 2451, -2450, -441, 3773, -3772, -576, 5500, -5499, -729, 7686, -7685, -900, 10385, -10384, -1089, 13651, -13650, -1296, 17538, -17537

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform of A115142.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-10,-16,-19,-16,-10,-4,-1).

Cf. A000108, A099504, A115142, A128504.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 )); // G. C. Greubel, Jun 16 2022

LinearRecurrence[{-4,-10,-16,-19,-16,-10,-4,-1}, {1,-9,26,-25,-36,133,-132,-81}, 40] (* G. C. Greubel, Jun 16 2022 *)

def A144701_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)*(1-5*x+x^2)/(1+x+x^2)^4 ).list()
A144701_list(40) # G. C. Greubel, Jun 16 2022

G.f.: (1+x)*(1-x)*(1-5*x+x^2)/(1+x+x^2)^4.

a(n) = (6 - 7*n - 9*n^2 - 2*n^3)*cos(2*Pi*n/3)/6 - sqrt(3)*(42 + 55*n + 21*n^2 + 2*n^3)*sin(2*Pi*n/3)/18.

A238988 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, 0, 2, 1, -1, -1, 1, 2, 1, 0, -1, -2, 1, 3, 1, 1, 0, -4, -2, 3, 3, 1, 1, 1, -2, -4, -2, 3, 4, 1, 0, 1, 3, -2, -9, -2, 6, 4, 1, -1, 0, 6, 3, -9, -9, 0, 6, 5, 1, -1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1, 0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1

Offset: 0

Philippe Deléham, Mar 07 2014

T(n,0) = T(n+1,1) = A010892(n), T(n+2,2) = T(n+3,3) = A099254(n), T(n+4,4) = T(n+5,5) = A128504(n).

Triangle T(n,k) = A101950(n - floor((k+1)/2),floor(k/2)).

			Triangle begins:
1;
1, 1;
0, 1, 1;
-1, 0, 2, 1;
-1, -1, 1, 2, 1;
0, -1, -2, 1, 3, 1;
1, 0, -4, -2, 3, 3, 1;
1, 1, -2, -4, -2, 3, 4, 1;
0, 1, 3, -2, -9, -2, 6, 4, 1;
-1, 0, 6, 3, -9, -9, 0, 6, 5, 1;
-1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1;
0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1;
1, 0, -8, -4, 18, 18, -6, -24, -20, 5, 15, 6, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A101950

nmax=11; Flatten[CoefficientList[Series[CoefficientList[Series[(1 + x*y)/(1 - x + x^2 - x^2*y^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

G.f.: (1 + x*y)/(1 - x + x^2 - x^2*y^2).
T(n,k) = T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A010892(n), A040000(n), A105476(n+1) for x = -1, 0, 1, 2 respectively.

A281862 Riordan transform of the triangular number sequence A000217 with the Chebyshev S matrix A049310.

Original entry on oeis.org

0, 1, 3, 4, 1, -6, -11, -6, 9, 21, 14, -12, -34, -25, 15, 50, 39, -18, -69, -56, 21, 91, 76, -24, -116, -99, 27, 144, 125, -30, -175, -154, 33, 209, 186, -36, -246, -221, 39, 286, 259, -42, -329, -300, 45, 375

Offset: 0

Wolfdieter Lang, Feb 18 2017

For the analogous sequence with the inverse S Riordan matrix A053121 see A189391.

Cf. A000217, A049310, A128504, A189391.

CoefficientList[Series[x (1 + x^2)/(1 - x + x^2)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = Sum_{m=0..n} A049310(n,m)*A000217(m), n >= 0.
a(n) = b(n-1) + b(n-3), n >= 0 with b(-3) = b(-2) = b(-1) = 0 and  b(n) = A128504(n) for n >= 0.
G.f.: (1/(1+x^2))*Tri(x/(1+x^2)), with Tri(x) = x/(1-x)^3 (g.f. of A000217).
G.f. x*(1 + x^2)/(1 - x + x^2)^3.

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A101950 Product of A049310 and A007318 as lower triangular matrices.

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A128503 Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.

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A147621 The 3rd Witt transform of A000292.

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A147618 The 3rd Witt transform of A000217.

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A144701 Hankel transform of expansion of 1/c(x)^3, c(x) the g.f. of A000108.

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A238988 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A281862 Riordan transform of the triangular number sequence A000217 with the Chebyshev S matrix A049310.

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