A159854 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(4 + 5\,x + 2\,x^{2} + n + 2\,n x\right)\,P_{n}(x) = \left(10 + 24\,x + 18\,x^{2} + 4\,x^{3} + 2\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 5 + 12\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	5	12	4								21
1	12	42	40	8							102
2	28	128	196	112	16						480
3	64	360	752	712	288	32					2208
4	144	960	2528	3328	2256	704	64				9984
5	320	2464	7808	13120	12480	6560	1664	128			44544
6	704	6144	22720	46336	56640	41984	17984	3840	256		196608
7	1536	14976	63232	151424	225792	215936	130816	47232	8704	512	860160

All 15 rows

Technical details

Node id: n-23ddee92588e3600 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b6b17fe263ea2de9

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:    5,    12,      4;
  1:   12,    42,     40,       8;
  2:   28,   128,    196,     112,      16;
  3:   64,   360,    752,     712,     288,      32;
  4:  144,   960,   2528,    3328,    2256,     704,      64;
  5:  320,  2464,   7808,   13120,   12480,    6560,    1664,     128;
  6:  704,  6144,  22720,   46336,   56640,   41984,   17984,    3840,     256;
  7: 1536, 14976,  63232,  151424,  225792,  215936,  130816,   47232,    8704,    512;
  8: 3328, 35840, 169984,  466944,  820736,  960512,  752640,  385024,  120064,  19456,  1024;
  9: 7168, 84480, 444416, 1376256, 2783232, 3849216, 3698688, 2451456, 1084416, 297472, 43008, 2048;

%F Formula

The row polynomials satisfy (n+4+(2*n+5)*x+2*x^2)*P(n,x) = (2*n+10+(6*n+24)*x+(4*n+18)*x^2+4*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 5 + 12*x + 4*x^2.

%t Mathematica

P[0, x_] := 5 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 10 + (6*n + 24)*x + (4*n + 18)*x^2 + 4*x^3)*P[n - 1, x])/(n + 4 + (2*n + 5)*x + 2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 5 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 10 + (6*n + 24)*x + (4*n + 18)*x^2 + 4*x^3)*v[i - 1])/(n + 4 + (2*n + 5)*x + 2*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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A391533 → reverse → D+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.