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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	3328	35840	169984	466944	820736	960512	752640	385024	120064	19456	1024							3735552
9	7168	84480	444416	1376256	2783232	3849216	3698688	2451456	1084416	297472	43008	2048						16121856
10	15360	196608	1135616	3911680	8939520	14254080	16214016	13197312	7572480	2949120	721920	94208	4096					69206016
11	32768	452608	2846720	10790912	27484160	49606656	65150976	62853120	44335104	22415360	7794688	1722368	204800	8192				295698432
12	69632	1032192	7020544	29032448	81506304	164184064	244383744	272498688	228028416	141836288	64069632	20119552	4050944	442368	16384			1258291200
13	147456	2334720	17072128	76464128	234504192	521297920	866877440	1096482816	1061339136	783704064	435781632	177848320	50905088	9412608	950272	32768		5335154688
14	311296	5242880	41009152	197722112	657506304	1598291968	2935668736	4151640064	4561649664	3898605568	2574860288	1294139392	481574912	126615552	21643264	2031616	65536	22548578304

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.