A272867 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 36\,x + 76\,x^{2} + 68\,x^{3} + 22\,x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	24	22													52
2	36	288	840	1056	484											2704
3	216	2592	12744	32832	46728	34848	10648									140608
4	1296	20736	143424	559872	1348704	2052864	1928256	1022208	234256							7311616
5	7776	155520	1386720	7257600	24684480	57003264	90509760	97574400	68360160	28110720	5153632					380204032
6	46656	1119744	12223872	80248320	352797120	1094114304	2454195456	4011752448	4743161280	3955944960	2209502592	742123008	113379904			19770609664
7	279936	7838208	101243520	799497216	4311667584	16797279744	48745452672	107041720320	178733326464	225830094336	212549242752	144511589376	67100288640	19047823872	2494357888	1028071702528

All 15 rows

Technical details

Node id: n-25060c57f9abbca7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-38e8cf17451719bf

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        6,        24,         22;
  2:       36,       288,        840,        1056,          484;
  3:      216,      2592,      12744,       32832,        46728,         34848,          10648;
  4:     1296,     20736,     143424,      559872,      1348704,       2052864,        1928256,        1022208,         234256;
  5:     7776,    155520,    1386720,     7257600,     24684480,      57003264,       90509760,       97574400,       68360160,        28110720,         5153632;
  6:    46656,   1119744,   12223872,    80248320,    352797120,    1094114304,     2454195456,     4011752448,     4743161280,      3955944960,      2209502592,       742123008,       113379904;
  7:   279936,   7838208,  101243520,   799497216,   4311667584,   16797279744,    48745452672,   107041720320,   178733326464,    225830094336,    212549242752,    144511589376,     67100288640,     19047823872,      2494357888;
  8:  1679616,  53747712,  801736704,  7399268352,  47285296128,  221852639232,   790464116736,  2181681340416,  4713801205248,   7999498248192,  10627350902784,  10936513437696,   8546963217408,   4903948836864,   1948320270336,    478916714496,    54875873536;
  9: 10077696, 362797056, 6137316864, 64819740672, 478932424704, 2628746846208, 11107524556800, 36941984907264, 98033369969664, 209125207904256, 359455689888768, 496664463753216, 547559821670400, 475154105868288, 317417884489728, 157520061186048, 54686302336512, 11853188683776, 1207269217792;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+36*x+76*x^2+68*x^3+22*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 36*x + 76*x^2 + 68*x^3 + 22*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 36*x + 76*x^2 + 68*x^3 + 22*x^4)*v[i - 1])/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272867 → basis A013610 → x ↦ x+1

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.