A156289 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x) + \left(5 + 7\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	31	45	15													91
4	379	798	525	105												1807
5	6556	17895	17325	6930	945											49651
6	150349	500643	634425	377685	103950	10395										1777447
7	4373461	17106180	26636610	20930910	8648640	1756755	135135									79587691
8	156297964	699786363	1287181350	1250989740	687161475	210945735	33108075	2027025								4327497727
9	6698486371	33693021225	71265219495	82244970990	56198727585	23069436390	5499724230	689188500	34459425							279393234211
10	337789490599	1881901591458	4488768443775	5982583378695	4877453844900	2503203848145	801824873850	153206603550	15713497800	654729075						21043100301847
11	19738202807236	120452544462855	319244074196355	481553509984470	455214215928780	279939935330055	112591882137735	29017592604000	4551021800325	389563799625	13749310575					1822706292362011
12	1319703681935929	8742930696168903	25451659566967425	42791091536360475	45937259351312175	32875820841453975	15904152942452775	5165353436668425	1096082599186575	143781123786300	10435726726425	316234143225				179438587737162607
13	99896787342523081	713201131859372520	2259354428517671940	4183441282882761900	5020412178253071600	4093301891671042050	2312158979083303350	906227666609265000	243070437729994875	43246148493673125	4817827172032875	300422436063750	7905853580625			19879437087904356691
14	8484301665702298804	64876512509024645523	221950955006521731900	448266334445545313880	593978499720196841250	542885215684461700050	350390043841436849250	160834442240633411250	52209167619824089875	11770699675025040750	1781952517170700875	170752206805054875	9249848689331250	213458046676875		2457608340439083686407
15	802221679220975886631	6536005919360954127045	23994334689347643435375	52442671354235451869580	75968691332431684820670	76854141742141325261700	55750163149192727615400	29318406525296791402500	11175486927115458197250	3056733261302955609375	587602742889902614875	76628829825316518750	6384245565376428750	303323884327839375	6190283353629375	336569781912094245256651

Technical details

Node id: n-992ded70fabcb6bc · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-27428437fc285c33

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:            4,             3;
   3:           31,            45,            15;
   4:          379,           798,           525,           105;
   5:         6556,         17895,         17325,          6930,           945;
   6:       150349,        500643,        634425,        377685,        103950,         10395;
   7:      4373461,      17106180,      26636610,      20930910,       8648640,       1756755,       135135;
   8:    156297964,     699786363,    1287181350,    1250989740,     687161475,     210945735,     33108075,      2027025;
   9:   6698486371,   33693021225,   71265219495,   82244970990,   56198727585,   23069436390,   5499724230,    689188500,    34459425;
  10: 337789490599, 1881901591458, 4488768443775, 5982583378695, 4877453844900, 2503203848145, 801824873850, 153206603550, 15713497800, 654729075;

%F Formula

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + (5+7*x+2*x^2)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 3*x)*P[n - 1, x] + (5 + 7*x + 2*x^2)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4 + 3*x)*v[i - 1] + (5 + 7*x + 2*x^2)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.