A185962 -id:A185962 - cp's OEIS frontend

A185963 Row sums of number triangle A185962.

1, 0, -2, -3, 0, 7, 11, 1, -24, -40, -7, 82, 145, 37, -279, -524, -174, 945, 1888, 767, -3185, -6783, -3244, 10676, 24301, 13330, -35567, -86823, -53615, 117672, 309366, 212101, -386224, -1099385, -827997, 1255937, 3896480, 3197152, -4039199, -13773374

Offset: 0

Paul Barry, Feb 07 2011

			G.f. = 1 - 2*x^2 - 3*x^3 + 7*x^5 + 11*x^6 + x^7 - 24*x^8 - 40*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (2,-3,1).

Cf. A000931.

a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27):
seq(simplify(a(n)), n=0..39); # Peter Luschny, Nov 03 2017

LinearRecurrence[{2,-3,1},{1,0,-2},50] (* Vincenzo Librandi, Feb 18 2012 *)

x='x+O('x^50); Vec((1-x)^2/(1-2*x+3*x^2-x^3)) \\ G. C. Greubel, Jul 23 2017

G.f.: (1-x)^2/(1-2x+3x^2-x^3).
a(n) = Sum_{k=0..n} Sum_{i=0..(2k+2)} C(2k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).
a(n) = Sum_{k=0..n} binomial(n+2k,3k)*(-1)^k = Sum_{k=0..n} A109955(n,k)*(-1)^k. - Philippe Deléham, Feb 18 2012
a(n) = A000931(-3*n). - Michael Somos, Sep 18 2012
a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27). - Peter Luschny, Nov 03 2017

More terms from Philippe Deléham, Feb 07 2012

A185964 Diagonal sums of number triangle A185962.

Original entry on oeis.org

1, -1, 0, -2, 1, 0, 4, 0, 1, -7, -3, -5, 10, 9, 16, -9, -17, -40, -6, 19, 82, 54, 10, -135, -161, -127, 153, 340, 433, 0, -527, -1053, -620, 434, 2013, 2200, 712, -2880, -5267, -4491, 1981, 9635, 13350, 4897, -12392

Offset: 0

Paul Barry, Feb 07 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-2,1).

Cf. A185962.

CoefficientList[Series[(1 - x)^2/(1 - x + 2*x^3 - x^4), {x, 0, 50}], x] (* G. C. Greubel, Jul 23 2017 *)

x='x+O('x^50); Vec((1 - x)^2/(1 - x + 2*x^3 - x^4)) \\ G. C. Greubel, Jul 23 2017

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

a(n) = Sum_{k=0..floor(n/2)} A185962(n-k,k).

More terms from Philippe Deléham, Feb 07 2012

A185965 Central coefficients of number triangle A185962.

Original entry on oeis.org

1, -2, 0, 8, -10, -30, 98, 40, -648, 680, 3058, -8712, -6760, 65674, -52710, -348128, 856358, 1011330, -7116754, 3891920, 41214978, -87043088, -143941360, 793389048, -224365750, -4961373872, 8914590594, 19893652520, -89559777800, 540262170, 601349934194, -905363401312, -2693832315240, 10150582469480, 2943320005570, -73015796693016, 89846661676688

Offset: 0

Paul Barry, Feb 07 2011

a(n)=A185962(2n,n); a(n)=sum{i=0..2n+2, C(2n+2,i)*sum{C(n+j,j)*C(j,n-i-j)*(-1)^(n-j)}}.
Conjecture: 15*n*(n-1)*a(n) +10*(2*n-1)*(n-1)*a(n-1) +(109*n^2-263*n+180)*a(n-2) +4*(-2*n^2-10*n+57)*a(n-3) +60*(n-4)^2*a(n-4) -6*(2*n-7)*(n-5)*a(n-5)=0. - R. J. Mathar, Dec 03 2014
Conjecture: 3*n*(n-1)*(57*n-136)*a(n) +(n-1)*(171*n^2-415*n+14)*a(n-1) +2*(532*n^3-2281*n^2+2755*n-840)*a(n-2) -2*(n-3)*(304*n^2-835*n+216)*a(n-3) +2*(19*n-5)*(n-4)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Dec 03 2014

A195350 Expansion of (1 - 3x - x^2)/(1 - 4x + 2*x^3 + x^4).

Original entry on oeis.org

1, 1, 3, 10, 37, 141, 541, 2080, 8001, 30781, 118423, 455610, 1752877, 6743881, 25945881, 99822160, 384048001, 1477556361, 5684635243, 21870622810, 84143330517, 323726495221, 1245480100021, 4791763116240, 18435456144001, 70927137880741

Offset: 0

Bruno Berselli, Sep 16 2011

Rewrite the Girard-Waring formulae to express the mean powers in terms of the mean symmetric functions of the data values; the results are polynomials in the mean symmetric polynomials, indexed by the power n. Then for 3 data points, the sum of the positive coefficients in the n-th such polynomial is a(n). a(n+1)/a(n) approaches 1/(2^(1/3)-1). See extended comment in A301417. - Gregory Gerard Wojnar, Mar 19 2018

Bruno Berselli, Table of n, a(n) for n = 0..1000
G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See Table GW. n=3 p. 22.
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1).

Cf. A185962 (gives the coefficients of numerator and denominator of the g.f., row 4 and 5 of its triangular array). Sequences likewise related to A185962: A000012 (row 1 and 2), A001333 (row 2 and 3) and A006190 (row 3 and 4).

Cf. also A195339, A301417, A301420, A301421, A301424, A302764, A301483, A303647.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x-x^2)/(1-4*x+2*x^3+x^4)));

[seq(coeftayl((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018

CoefficientList[Series[(1 - 3 x - x^2)/(1 - 4 x + 2 x^3 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 26 2013 *)

makelist(coeff(taylor((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x, 0, n), x, n), n, 0, 25);

Vec((1-3*x-x^2)/(1-4*x+2*x^3+x^4)+O(x^26))

G.f.: (1-3*x-x^2)/((1-x)*(1-3*x-3*x^2-x^3)).
a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4).
a(n) = A301483(n) - A303647(n-2) + A195339(n-4) (conjectured). - Gregory Gerard Wojnar, Apr 27 2018

A195339 Expansion of 1/(1-4x+2x^3+x^4).

Original entry on oeis.org

1, 4, 16, 62, 239, 920, 3540, 13620, 52401, 201604, 775636, 2984122, 11480879, 44170640, 169938680, 653808840, 2515413201, 9677604804, 37232862856, 143246816182, 551116641919, 2120323237160, 8157566453420, 31384785713660, 120747379738401

Offset: 0

Bruno Berselli, Sep 16 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1).

Cf. A185962 (gives the coefficients of the denominator of the g.f., row 5 of its triangular array). Sequences likewise related to A185962: A000007 (row 1), A000012 (row 2), A000129 (row 3) and A006190 (row 4).

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-4*x+2*x^3+x^4)));

CoefficientList[Series[1/(1-4x+2x^3+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{4,0,-2,-1},{1,4,16,62},30] (* Harvey P. Dale, Dec 02 2011 *)

makelist(coeff(taylor(1/(1-4*x+2*x^3+x^4), x, 0, n), x, n), n, 0, 24);

PARI
```
Vec(1/(1-4*x+2*x^3+x^4)+O(x^25))
```

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

a(n) = 4*a(n-1)-2*a(n-3)-a(n-4).

A185966 Series reversion of A028310.

Original entry on oeis.org

1, -1, 0, 2, -2, -5, 14, 5, -72, 68, 278, -726, -520, 4691, -3514, -21758, 50374, 56185, -374566, 194596, 1962618, -3956504, -6258320, 33057877, -8974630, -190822072, 330170022, 710487590, -3088268200, 18008739, 19398384974, -28292606291, -81631282280, 298546543220, 84094857302, -2028216574806, 2428288153424, 9450205225145

Offset: 0

Paul Barry, Feb 07 2011

			1 - x + 2*x^3 - 2*x^4 - 5*x^5 + 14*x^6 + 5*x^7 - 72*x^8 + 68*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028130, A185962, A185965.

CoefficientList[1/x*InverseSeries[Series[x*(1-x+x^2) /(1-x)^2, {x, 0, 20}], x],x] (* Vaclav Kotesovec, Jan 22 2014 *)

{a(n) = if( n<0, 0, polcoeff( serreverse( x * ((1 - x + x^2) / (1 - x)^2 + x * O(x^n))) / x, n))} /* Michael Somos, Apr 05 2012 */

{a(n) = local(B); if( n<0, 0, B = O(x); for( k=0, n, B = (1 - B) * (x + B * (B - x))); polcoeff( B / x, n))} /* Michael Somos, Apr 05 2012 */

a(n) = A185962(2*n,n)/(n+1) = A185965(n)/(n+1).
Given g.f. A(x) then B(x) = x * A(x) satisfies B(x) = (1 - B(x)) * (x + B(x) * (B(x) - x)). - Michael Somos, Apr 05 2012
Conjecture: 6*n*(n+1)*a(n) -n*(n-14)*a(n-1) +2*n*(14*n-19)*a(n-2) -4*(n-2)*(17*n-48)*a(n-3) +6*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2012
Recurrence (of order 3): 3*n*(n+1)*(19*n-27)*a(n) = -2*n*(38*n^2 - 73*n + 9)*a(n-1) - 20*(19*n^3 - 65*n^2 + 66*n - 18)*a(n-2) + 2*(n-3)*(2*n-3)*(19*n-8)*a(n-3). - Vaclav Kotesovec, Jan 22 2014
Lim sup n->infinity |a(n)|^(1/n) = sqrt(20/9 + 1/27*(272376 - 12312 * sqrt(57))^(1/3) + 2/9*(1261 + 57 * sqrt(57))^(1/3)) = 2.637962913244886521522... - Vaclav Kotesovec, Jan 22 2014

A185967 Inverse of Riordan array ((1-x)(1-x^2)(1-x^3)/(1-x^6), x(1-x)(1-x^2)(1-x^3)/(1-x^6)).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 37, 26, 12, 4, 1, 146, 103, 49, 18, 5, 1, 602, 426, 207, 80, 25, 6, 1, 2563, 1818, 897, 359, 120, 33, 7, 1, 11181, 7946, 3966, 1628, 570, 170, 42, 8, 1, 49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1, 224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1

Offset: 0

Paul Barry, Feb 07 2011

Riordan array (g(x),xg(x)) where x*g(x) = (x+2)/3 - 2*sqrt(1+x+x^2) * cos(arccos(-(2x^3+3x^2+24x-2) / (2(1+x+x^2)^(3/2)))/3)/3.

			Triangle begins
  1,
  1, 1,
  3, 2, 1,
  10, 7, 3, 1,
  37, 26, 12, 4, 1,
  146, 103, 49, 18, 5, 1,
  602, 426, 207, 80, 25, 6, 1,
  2563, 1818, 897, 359, 120, 33, 7, 1,
  11181, 7946, 3966, 1628, 570, 170, 42, 8, 1,
  49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1,
  224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1
Production matrix is
  1, 1,
  2, 1, 1,
  3, 2, 1, 1,
  4, 3, 2, 1, 1,
  5, 4, 3, 2, 1, 1,
  6, 5, 4, 3, 2, 1, 1,
  7, 6, 5, 4, 3, 2, 1, 1,
  8, 7, 6, 5, 4, 3, 2, 1, 1,
  9, 8, 7, 6, 5, 4, 3, 2, 1, 1
  10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See p. 22.

Inverse of number triangle A185962.

First column is A109081. Row sums are A106228(n+1).

T := (n, k) -> `if`(n=k, 1, (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4)):
seq(seq(simplify(T(n, k)), k=0..n),n=0..10); # Peter Luschny, Apr 02 2019

T[n_, k_] := (k+1)/(n+1) Sum[Binomial[n+1, i] Binomial[n-k+i-1, n-k-i], {i, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 25 2019, after Vladimir Kruchinin *)

T(n,k):=(k+1)/(n+1)*sum(binomial(n+1,i)*binomial(n-k+i-1,n-k-i),i,0,n-k); /* Vladimir Kruchinin, Apr 02 2019 */

T(n, k) = (k + 1)/(n + 1)*Sum_{i=0..n-k} C(n+1, i)*C(n-k+i-1, n-k-i). - Vladimir Kruchinin, Apr 02 2019

T(n, k) = (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4) for k < n. - Peter Luschny, Apr 02 2019

cp's OEIS Frontend

A185963 Row sums of number triangle A185962.

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A185964 Diagonal sums of number triangle A185962.

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A185965 Central coefficients of number triangle A185962.

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A195350 Expansion of (1 - 3*x - x^2)/(1 - 4*x + 2*x^3 + x^4).

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A195339 Expansion of 1/(1-4*x+2*x^3+x^4).

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A185966 Series reversion of A028310.

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A185967 Inverse of Riordan array ((1-x)(1-x^2)(1-x^3)/(1-x^6), x(1-x)(1-x^2)(1-x^3)/(1-x^6)).

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A195350 Expansion of (1 - 3x - x^2)/(1 - 4x + 2*x^3 + x^4).

A195339 Expansion of 1/(1-4x+2x^3+x^4).