A103633 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 26\,x + 20\,x^{2} + 12\,n x + 29\,n x^{2} + 4\,n x^{3} - 16\,n x^{4} + 9\,n^{2} x^{2} + 24\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(32 + 172\,x + 304\,x^{2} + 148\,x^{3} - 88\,x^{4} - 64\,x^{5} - 24\,n - 48\,n x + 128\,n x^{2} + 402\,n x^{3} + 296\,n x^{4} + 32\,n x^{5} - 54\,n^{2} x - 195\,n^{2} x^{2} - 182\,n^{2} x^{3} + 48\,n^{2} x^{4} + 96\,n^{2} x^{5} - 27\,n^{3} x^{2} - 108\,n^{3} x^{3} - 144\,n^{3} x^{4} - 64\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(8 + 52\,x + 116\,x^{2} + 80\,x^{3} - 56\,x^{4} - 96\,x^{5} - 32\,x^{6} + 24\,n x + 146\,n x^{2} + 326\,n x^{3} + 316\,n x^{4} + 112\,n x^{5} + 18\,n^{2} x^{2} + 102\,n^{2} x^{3} + 212\,n^{2} x^{4} + 192\,n^{2} x^{5} + 64\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	2	12	12													26
3	2	20	48	32												102
4	4	48	156	192	80											480
5	4	64	300	608	560	192										1728
6	8	144	792	2016	2640	1728	448									7776
7	8	176	1224	4128	7680	8064	4480	1024								26784
8	16	384	2976	11520	25680	34560	27776	12288	2304							117504
9	16	448	4128	19456	54480	96192	108416	75776	29952	5120						393984
10	32	960	9600	49920	157600	323136	441280	399360	230400	76800	11264					1700352
11	32	1088	12480	75520	282400	701376	1195264	1408000	1128960	588800	180224	24576				5598720
12	64	2304	28224	184320	753600	2080512	4027072	5548032	5425920	3686400	1655808	442368	53248			23887872
13	64	2560	35136	259584	1214400	3889152	8881600	14752768	17908992	15744000	9765888	4055040	1011712	114688		77635584
14	128	5376	77952	612864	3077760	10692864	26837888	49806336	69011712	71285760	54168576	29417472	10809344	2408448	245760	328458240

Technical details

Node id: n-75ad3144101df727 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7f8d68687ddb27c

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 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:  1,   4;
  2:  2,  12,   12;
  3:  2,  20,   48,    32;
  4:  4,  48,  156,   192,    80;
  5:  4,  64,  300,   608,   560,   192;
  6:  8, 144,  792,  2016,  2640,  1728,    448;
  7:  8, 176, 1224,  4128,  7680,  8064,   4480,  1024;
  8: 16, 384, 2976, 11520, 25680, 34560,  27776, 12288,  2304;
  9: 16, 448, 4128, 19456, 54480, 96192, 108416, 75776, 29952, 5120;

%F Formula

The row polynomials satisfy (8+(12*n+26)*x+(9*n^2+29*n+20)*x^2+(24*n^2+4*n)*x^3+(16*n^2-16*n)*x^4)*P(n,x) = -(24*n-32+(54*n^2+48*n-172)*x+(27*n^3+195*n^2-128*n-304)*x^2+(108*n^3+182*n^2-402*n-148)*x^3+(144*n^3-48*n^2-296*n+88)*x^4+(64*n^3-96*n^2-32*n+64)*x^5)*P(n-1,x) + (8+(24*n+52)*x+(18*n^2+146*n+116)*x^2+(102*n^2+326*n+80)*x^3+(212*n^2+316*n-56)*x^4+(192*n^2+112*n-96)*x^5+(64*n^2-32)*x^6)*d/dx 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[(-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*P[n - 1, x] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*D[P[n - 1, x], x])/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*n)*x^4)]];
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] = (-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*v[i - 1] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*deriv(v[i - 1], x))/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*n)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A064861 → θ+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.