A024537 -id:A024537 - cp's OEIS frontend

A301417 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 4 data.

1, 4, 19, 98, 516, 2725, 14400, 76105, 402229, 2125864, 11235643, 59382770, 313850616, 1658767513, 8766940464, 46335152161, 244891172089, 1294302130684, 6840663104371, 36154365042098, 191083538489436, 1009917298758493, 5337628549243344, 28210506508524169

Offset: 1

Gregory Gerard Wojnar, Mar 20 2018

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 4 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/4)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.)

More precisely, given a finite collection X:=(x(i), i =1..n) of data, the Girard-Waring formulae express the sum of the k-th powers of the data, S_k(X):=Sum(x(i)^k, i=1..n), in terms of the elementary symmetric polynomials in the data. The j-th elementary symmetric polynomial is s_j(X):=Sum(Product(x(i), x(i) in X_0), X_0 \subseteq X, where |X_0|=j). So the Girard-Waring formulae provide coefficients a(J,k) such that S_k(X)=Sum(a(J,k)*Product(s_j(X), j \in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). [Thus J is an integer partition of k.] By "mean powers" I mean T_k(X):=Sum(x(i)^k, i=1..n)/n. By the "mean symmetric polynomials" I mean t_j(X):=s_j(X)/binomial(n,j). The Girard-Waring mean formulae then provide coefficients b(J,k,n) such that T_k(X)=Sum(b(J,k,n)*Product(t_j(X), j in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). So the sums of positive coefficients that I reference, for a fixed data set size n, and a fixed power k, are Sum(b(J,k,n), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k, such that b(J,k,n)>0).

Gregory Gerard Wojnar, Table of n, a(n) for n = 1..68
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=4 p. 23.
Gregory Gerard Wojnar, Java program. Within the program, the variable I denotes the number of data; J denotes the exponent.
Michel Marcus, pari script (translated from java)
Index entries for linear recurrences with constant coefficients, signature (5, 2, -2, -3, -1).

Cf. A302764, A024537, A195350, A301420, A301421, A301424.

CoefficientList[Series[(-x (x + 1)^3 + 1)/(x^5 + 3 x^4 + 2 x^3 - 2 x^2 - 5 x + 1), {x, 0, 23}], x] (* Michael De Vlieger, Apr 07 2018 *)
LinearRecurrence[{5, 2, -2, -3, -1}, {1, 4, 19, 98, 516}, 24] (* Jean-François Alcover, Dec 02 2018 *)

lista(4, nn) \\ use pari script link;  Michel Marcus, Apr 21 2018

G.f.: (-x*(x+1)^3+1)/(x^5+3*x^4+2*x^3-2*x^2-5*x+1); this denominator equals (1-x)*(2-(1+x)^4).

a(n+5) = 5*a(n+4)+2*a(n+3)-2*a(n+2)-3*a(n+1)-a(n).

A301420 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 data.

Original entry on oeis.org

1, 5, 31, 205, 1376, 9251, 62210, 418361, 2813485, 18920751, 127242501, 855708865, 5754662616, 38700243965, 260260067876, 1750255192001, 11770508100345, 79156948982921, 532332378421395, 3579947998967501, 24075236064574376

Offset: 1

Gregory Gerard Wojnar, Mar 20 2018

nonn

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 5 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/5)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Gregory Gerard Wojnar, Table of n, a(n) for n = 1..65
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=5 p. 23.

Cf. A302764, A024537, A195350, A301417, A301421, A301424.

lista(5, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

G.f.: (-x*(x+1)^4+1)/(x^6+4*x^5+5*x^4-5*x^2-6*x+1); this denominator equals (1-x)*(2-(x+1)^5) (conjectured).

a(n+14) = 7*a(n+13) - a(n+12) - 6*a(n+11) + 2*a(n+10) - a(n+9) + 4*a(n+8) + a(n+7) + 4*a(n+5) + 2*a(n+4) - a(n+3) - 5*a(n+2) - 4*a(n+1) - a(n) (conjectured).

A301421 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

Original entry on oeis.org

1, 6, 46, 371, 3026, 24707, 201748, 1647429, 13452565, 109850886, 897019828, 7324880157, 59813470848, 488424550081, 3988374821616, 32568251770049, 265945672309613, 2171657880797162, 17733313387923690, 144806604435722311, 1182461068019218530, 9655734852907204771

Offset: 1

Gregory Gerard Wojnar, Mar 20 2018

nonn

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 6 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/6)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Gregory Gerard Wojnar, Table of n, a(n) for n = 1..62 [a(21) corrected by _Georg Fischer_, Aug 18 2021]
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=6 p. 24.

Cf. A301764, A024537, A195350, A301417, A301420, A301424.

lista(6, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

G.f.: (-x*(x+1)^5+1)/(x^7+5*x^6+9*x^5+5*x^4-5*x^3-9*x^2-7*x+1); this denominator equals (1-x)*(2-(1+x)^6) (conjectured).

a(21) corrected by Georg Fischer, Aug 18 2021

A301424 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 7 data.

Original entry on oeis.org

1, 7, 64, 609, 5846, 56161, 539540, 5183417, 49797685, 478412117, 4596160548, 44155846113, 424210322004, 4075437640457, 39153200900024, 376149330687809, 3613710136705565, 34717331354145139, 333533418773956668, 3204294140706218329, 30784024515164777522

Offset: 1

Gregory Gerard Wojnar, Mar 20 2018

nonn

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 7 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/7)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) The sums of the negative coefficients are 1 less than the corresponding sums of the positive coefficients. See extended comment in A301417.

G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017.

Cf. A302764, A024537, A195350, A301417, A301420, A301421.

lista(7, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

G.f.: (-x*(x+1)^6+1)/(x^2*(x^6+6*x^5+14*x^4+14*x^3-14*x-14)-8*x+1); this denominator equals (1-x)*(2-(1+x)^7) (conjectured).

A302764 Pascal-like triangle with A000012 as the left border and A080956 as the right border.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 2, -2, 1, 4, 5, 0, -5, 1, 5, 9, 5, -5, -9, 1, 6, 14, 14, 0, -14, -14, 1, 7, 20, 28, 14, -14, -28, -20, 1, 8, 27, 48, 42, 0, -42, -48, -27, 1, 9, 35, 75, 90, 42, -42, -90, -75, -35, 1, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44

Offset: 1

Gregory Gerard Wojnar, Apr 12 2018

Number the rows of the triangle beginning with n=0. For each row construct a degree n polynomial with regularly decreasing powers, denoting the polynomial as f_n(x); e.g., for row 2 we have f_2(x)=1x^2+2x+0. Then construct g_n(x)=x^2*f_{n-1}(x)-(n+1)x+1. It obtains that g_n(x)=(1-x)(2-(1+x)^n). These g_n(x) are the denominators of the generating functions for the following sequences: A024537 (n=2); A195350 (n=3); A301417 (n=4); A301420 (n=5); A301421 (n=6); A301424 (n=7). For these sequences the asymptotic term-to-term ratios are 1/(2^(1/n)-1). The numerators of the generating functions are 1-x(x+1)^(n-1).

			Triangle begins:
  1;
  1, 1;
  1, 2,  0;
  1, 3,  2, -2;
  1, 4,  5,  0, -5;
  1, 5,  9,  5, -5,  -9;
  1, 6, 14, 14,  0, -14, -14;
  1, 7, 20, 28, 14, -14, -28, -20;
  ...

Cf. A000012, A007318, A024537, A080956, A195350, A301417, A301420, A301421, A301424.

T(n,k) = if (k==0, 1, if (k==n, (n+1)*(2-n)/2, if (k>n, 0, T(n-1,k) + T(n-1,k-1))));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 21 2018

T(n,k) = T(n-1,k) + T(n-1,k-1) with T(n, 0) = 1 and T(n, n) = (n+1)*(2-n)/2.

A113225 a(2n) = A011900(n), a(2n+1) = A001109(n+1).

Original entry on oeis.org

1, 1, 3, 6, 15, 35, 85, 204, 493, 1189, 2871, 6930, 16731, 40391, 97513, 235416, 568345, 1372105, 3312555, 7997214, 19306983, 46611179, 112529341, 271669860, 655869061, 1583407981, 3822685023, 9228778026, 22280241075, 53789260175

Offset: 0

Creighton Dement, Oct 18 2005

a(n+1) - a(n) = A097075(n+1), a(n) + a(n+1) = A024537(n+1), a(n+2) - a(n+1) - a(n) = A105635(n+1).
For n >= 1, a(n) is also the edge cover number and edge cut count of the n-Pell graph. - Eric W. Weisstein, Aug 01 2023
Also the independence number, Lovasz number, and Shannon capacity of the n-Pell graph. - Eric W. Weisstein, Aug 01 2023
Floretion Algebra Multiplication Program, FAMP Code: -2jbasejseq[B*C], B = - .5'i + .5'j - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; C = + .5'i + .5i' + .5'ii' + .5e

C. Dement, Floretion Integer Sequences (work in progress).

Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 19.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Edge Cut.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Lovasz Number.
Eric Weisstein's World of Mathematics, Pell Graph.
Eric Weisstein's World of Mathematics, Shannon Capacity.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A113224, A002315, A082639, A100828.

seq(iquo(fibonacci(n,2),1)-iquo(fibonacci(n,2),2),n=1..30); # Zerinvary Lajos, Apr 20 2008
with(combinat):seq(ceil(fibonacci(n,2)/2), n=1..30); # Zerinvary Lajos, Jan 12 2009

Ceiling[Fibonacci[Range[20], 2]/2]
Table[(1 + (-1)^n + 2 Fibonacci[n + 1, 2])/4, {n, 0, 20}] // Expand
CoefficientList[Series[-(-1 + x + x^2)/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x]
LinearRecurrence[{2, 2, -2, -1}, {1, 1, 3, 6}, 20]

{a(n)=local(y); if(n<0, 0, n++; y=x/(x^2+x-1)+x*O(x^n); polcoeff( y/(y^2-1), n))} /* Michael Somos, Sep 09 2006 */

G.f.: y/(y^2-1) where y=x/(x^2+x-1) if offset=1. - Michael Somos, Sep 09 2006
G.f.: (-1+x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)).
Diagonal sums of A119468. - Paul Barry, May 21 2006
a(n) = (1 + (-1)^n + 2 A000129(n+1))/4. - Eric W. Weisstein, Aug 01 2023
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Eric W. Weisstein, Aug 01 2023

A171842 Binomial transform of 1,0,1,0,2,0,4,0,8,0,16,...

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 57122, 137904, 332929, 803761, 1940450, 4684660, 11309769, 27304197, 65918162, 159140520, 384199201, 927538921, 2239277042, 5406093004, 13051463049, 31509019101, 76069501250, 183648021600, 443365544449, 1070379110497

Offset: 0

Philippe Deléham, Dec 19 2009

Number of nonisomorphic n-element interval orders with no 3-element antichain. - Richard Stanley, Nov 21 2011
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the number of Motzkin n-paths of height <= 2. - Alois P. Heinz, Nov 24 2023

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Michael D. Barrus, Weakly threshold graphs, arXiv preprint arXiv:1608.01358 [math.CO], 2016.
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv:1105.3713 [math.CO], 2011.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A001006.

read("transforms") :
L := [1,seq(2^i,i=0..30)] ;
AERATE(L,1) ;
BINOMIAL(%) ; # R. J. Mathar, Sep 26 2011

LinearRecurrence[{3, -1, -1}, {1, 1, 2}, 50] (* Jean-François Alcover, Feb 25 2017 *)

Vec((1-2*x)/((1-x)*(1-2*x-x^2)) + O(x^50)) \\ Colin Barker, Mar 16 2016

a(n) = A024537(n-1), n>0. - R. J. Mathar, Jan 28 2010
a(n) = 3*a(n-1)-a(n-2)-a(n-3). G.f.: (1-2*x)/((1-x)*(1-2*x-x^2)). - Colin Barker, Apr 01 2012
a(n) = (2+(1-sqrt(2))^n+(1+sqrt(2))^n)/4. - Colin Barker, Mar 16 2016

A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 2, 1, 4, 5, 3, 9, 12, 8, 21, 29, 20, 50, 70, 49, 120, 169, 119, 289, 408, 288, 697, 985, 696, 1682, 2378, 1681, 4060, 5741, 4059, 9801, 13860, 9800, 23661, 33461, 23660, 57122, 80782, 57121, 137904, 195025, 137903, 332929, 470832, 332928

Offset: 0

Clark Kimberling, Sep 12 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.
(row 0, the bottom row): A024537;
(row 1, the middle row): A000129;
(row 2, the top row): A048739;
(n-th column sum): A000129.

			First 10 columns:
  0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
  0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
  1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
  (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247309, A247310, A000129, A024537, A048739.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)

A301483 a(n) = floor(a(n-1)/(2^(1/3)-1)) with a(1)=1.

Original entry on oeis.org

1, 3, 11, 42, 161, 619, 2381, 9160, 35241, 135583, 521631, 2006882, 7721121, 29705639, 114287161, 439699520, 1691665681, 6508382763, 25039844851, 96336348522, 370636962881, 1425959779059, 5486126574341, 21106896023080, 81205027571321, 312421897357543

Offset: 1

Gregory Gerard Wojnar, Mar 22 2018

nonn

a(n+1)/a(n) approaches 1/(2^(1/3)-1).

Cf. A024537, A195350 (also has 1/(2^(1/3)-1) ratio), A303647.

[n le 1 select 1 else Floor(Self(n-1)/(2^(1/3)-1)): n in [1..30]]; // Vincenzo Librandi, Apr 04 2018

a:=proc(n) option remember;
   if n<1 then 0  else if n=1 then 1 else floor(a(n-1)/(2^(1/3)-1))
end if end if end proc:
seq(a(n), n=1..25);

RecurrenceTable[{a[1]==1, a[n]==Floor[a[n-1]/(2^(1/3)-1)]}, a, {n, 30}] (* Vincenzo Librandi, Apr 04 2018 *)

a=vector(50); a[1]=1; for(n=2, #a, a[n]=a[n-1]\(2^(1/3)-1)); a \\ Altug Alkan, Mar 22 2018

Conjectures from Colin Barker, Apr 01 2018: (Start)
G.f.: x*(1 - 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) for n>4.
(End)
a(n) = A195350(n) + A303647(n-2) - A195339(n-4) (conjectured).

A091186 Triangle read by rows, in which n-th row gives expansion of x^n/((1-x)(1-x-x^2)^n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 20, 38, 35, 19, 6, 1, 1, 33, 76, 86, 59, 26, 7, 1, 1, 54, 147, 197, 164, 91, 34, 8, 1, 1, 88, 277, 430, 420, 281, 132, 43, 9, 1, 1, 143, 512, 904, 1014, 792, 447, 183, 53, 10, 1, 1, 232, 932, 1846, 2338, 2087, 1371

Offset: 0

Paul Barry, Dec 25 2003

Riordan array (1/(1-x),x/(1-x-x^2)). - Paul Barry, Sep 13 2006

			Rows begin {1},{1,1},{1,2,1},{1,4,3,1}...

Row sums are A024537. Diagonal sums are A005578. Second column is A000071. Third column is A006478.

Essentially the vertical partial sums of triangle A037027.

G.f.: (1-y-y^2) / [(1-y(1+y+z))(1-y)].
Number triangle T(n,k)=sum{j=0..n-k, sum{i=0..n-k-j, C(k+j-1,j)C(j,n-k-i-j)}}; - Paul Barry, Sep 13 2006
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) - T(n-3,k), T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 20 2014

cp's OEIS Frontend

A301417 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 4 data.

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A301420 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 data.

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A301421 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

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A301424 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 7 data.

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A302764 Pascal-like triangle with A000012 as the left border and A080956 as the right border.

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A113225 a(2n) = A011900(n), a(2n+1) = A001109(n+1).

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A171842 Binomial transform of 1,0,1,0,2,0,4,0,8,0,16,...

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A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

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