A064861 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 + 40\,x + 50\,x^{2} + 24\,x^{3} + 4\,x^{4} + 31\,n + 62\,n x + 40\,n x^{2} + 8\,n x^{3} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-34 - 112\,x - 148\,x^{2} - 98\,x^{3} - 32\,x^{4} - 4\,x^{5} - 67\,n - 281\,n x - 404\,n x^{2} - 260\,n x^{3} - 76\,n x^{4} - 8\,n x^{5} - 96\,n^{2} - 265\,n^{2} x - 260\,n^{2} x^{2} - 108\,n^{2} x^{3} - 16\,n^{2} x^{4} - 27\,n^{3} - 54\,n^{3} x - 36\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(152 + 620\,x + 1016\,x^{2} + 852\,x^{3} + 384\,x^{4} + 88\,x^{5} + 8\,x^{6} + 156\,n + 530\,n x + 698\,n x^{2} + 444\,n x^{3} + 136\,n x^{4} + 16\,n x^{5} + 36\,n^{2} + 102\,n^{2} x + 106\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	5	5	1															11
1	12	18	8	1														39
2	16	38	31	10	1													96
3	36	104	113	57	13	1												324
4	44	168	255	195	78	15	1											756
5	96	416	744	708	384	117	18	1										2484
6	112	592	1328	1644	1221	552	146	20	1									5616
7	240	1392	3496	4976	4407	2505	906	198	23	1								18144
8	272	1840	5480	9440	10385	7593	3715	1190	235	25	1							40176
9	576	4192	13600	25920	32180	27254	16008	6485	1760	300	28	1						128304
10	640	5280	19632	43440	63600	64770	46915	24258	8835	2190	345	30	1					279936
11	1344	11776	46896	112272	180060	203976	167589	100849	44283	13935	3027	423	33	1				886464
12	1472	14336	63952	173040	316932	415128	400379	288263	155260	62013	17976	3633	476	35	1			1912896
13	3072	31488	148736	429184	845376	1202880	1275792	1025544	628736	293209	102774	26439	4788	567	38	1		6018624
14	3328	37376	194560	622720	1370656	2198112	2654176	2458520	1762825	979744	419524	136416	32830	5600	628	40	1	12877056

Technical details

Node id: n-9b991b5f89c05f75 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8ccb3246207975d6

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,    5,     1;
  1:  12,   18,     8,     1;
  2:  16,   38,    31,    10,     1;
  3:  36,  104,   113,    57,    13,     1;
  4:  44,  168,   255,   195,    78,    15,     1;
  5:  96,  416,   744,   708,   384,   117,    18,    1;
  6: 112,  592,  1328,  1644,  1221,   552,   146,   20,    1;
  7: 240, 1392,  3496,  4976,  4407,  2505,   906,  198,   23,   1;
  8: 272, 1840,  5480,  9440, 10385,  7593,  3715, 1190,  235,  25,  1;
  9: 576, 4192, 13600, 25920, 32180, 27254, 16008, 6485, 1760, 300, 28, 1;

%F Formula

The row polynomials satisfy (9*n^2+31*n+10+(12*n^2+62*n+40)*x+(4*n^2+40*n+50)*x^2+(8*n+24)*x^3+4*x^4)*P(n,x) = -(27*n^3+96*n^2+67*n+34+(54*n^3+265*n^2+281*n+112)*x+(36*n^3+260*n^2+404*n+148)*x^2+(8*n^3+108*n^2+260*n+98)*x^3+(16*n^2+76*n+32)*x^4+(8*n+4)*x^5)*P(n-1,x) + (36*n^2+156*n+152+(102*n^2+530*n+620)*x+(106*n^2+698*n+1016)*x^2+(48*n^2+444*n+852)*x^3+(8*n^2+136*n+384)*x^4+(16*n+88)*x^5+8*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 5 + 5*x + x^2.

%t Mathematica

P[0, x_] := 5 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 + 96*n^2 + 67*n + 34 + (54*n^3 + 265*n^2 + 281*n + 112)*x + (36*n^3 + 260*n^2 + 404*n + 148)*x^2 + (8*n^3 + 108*n^2 + 260*n + 98)*x^3 + (16*n^2 + 76*n + 32)*x^4 + (8*n + 4)*x^5)*P[n - 1, x] + (36*n^2 + 156*n + 152 + (102*n^2 + 530*n + 620)*x + (106*n^2 + 698*n + 1016)*x^2 + (48*n^2 + 444*n + 852)*x^3 + (8*n^2 + 136*n + 384)*x^4 + (16*n + 88)*x^5 + 8*x^6)*D[P[n - 1, x], x])/(9*n^2 + 31*n + 10 + (12*n^2 + 62*n + 40)*x + (4*n^2 + 40*n + 50)*x^2 + (8*n + 24)*x^3 + 4*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 5 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(27*n^3 + 96*n^2 + 67*n + 34 + (54*n^3 + 265*n^2 + 281*n + 112)*x + (36*n^3 + 260*n^2 + 404*n + 148)*x^2 + (8*n^3 + 108*n^2 + 260*n + 98)*x^3 + (16*n^2 + 76*n + 32)*x^4 + (8*n + 4)*x^5)*v[i - 1] + (36*n^2 + 156*n + 152 + (102*n^2 + 530*n + 620)*x + (106*n^2 + 698*n + 1016)*x^2 + (48*n^2 + 444*n + 852)*x^3 + (8*n^2 + 136*n + 384)*x^4 + (16*n + 88)*x^5 + 8*x^6)*deriv(v[i - 1], x))/(9*n^2 + 31*n + 10 + (12*n^2 + 62*n + 40)*x + (4*n^2 + 40*n + 50)*x^2 + (8*n + 24)*x^3 + 4*x^4)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
    
    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 reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
    no recorded property covers this requirement
    
    Parent-row provers of polytool polytool ? step › direct check

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.