A103633 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32 + 52\,x + 20\,x^{2} + 24\,n x + 29\,n x^{2} + 2\,n x^{3} - 4\,n x^{4} + 9\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(128 + 344\,x + 304\,x^{2} + 74\,x^{3} - 22\,x^{4} - 8\,x^{5} - 96\,n - 96\,n x + 128\,n x^{2} + 201\,n x^{3} + 74\,n x^{4} + 4\,n x^{5} - 108\,n^{2} x - 195\,n^{2} x^{2} - 91\,n^{2} x^{3} + 12\,n^{2} x^{4} + 12\,n^{2} x^{5} - 27\,n^{3} x^{2} - 54\,n^{3} x^{3} - 36\,n^{3} x^{4} - 8\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(64 + 208\,x + 232\,x^{2} + 80\,x^{3} - 28\,x^{4} - 24\,x^{5} - 4\,x^{6} + 96\,n x + 292\,n x^{2} + 326\,n x^{3} + 158\,n x^{4} + 28\,n x^{5} + 36\,n^{2} x^{2} + 102\,n^{2} x^{3} + 106\,n^{2} x^{4} + 48\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)

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

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	2														3
2	2	6	3													11
3	2	10	12	4												28
4	4	24	39	24	5											96
5	4	32	75	76	35	6										228
6	8	72	198	252	165	54	7									756
7	8	88	306	516	480	252	70	8								1728
8	16	192	744	1440	1605	1080	434	96	9							5616
9	16	224	1032	2432	3405	3006	1694	592	117	10						12528
10	32	480	2400	6240	9850	10098	6895	3120	900	150	11					40176
11	32	544	3120	9440	17650	21918	18676	11000	4410	1150	176	12				88128
12	64	1152	7056	23040	47100	65016	62923	43344	21195	7200	1617	216	13			279936
13	64	1280	8784	32448	75900	121536	138775	115256	69957	30750	9537	1980	247	14		606528
14	128	2688	19488	76608	192360	334152	419342	389112	269577	139230	52899	14364	2639	294	15	1912896

Technical details

Node id: n-8ca3f60c018179d4 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-91efc7c4f6d258e7

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,   2;
  2:  2,   6,    3;
  3:  2,  10,   12,    4;
  4:  4,  24,   39,   24,    5;
  5:  4,  32,   75,   76,   35,    6;
  6:  8,  72,  198,  252,  165,   54,    7;
  7:  8,  88,  306,  516,  480,  252,   70,   8;
  8: 16, 192,  744, 1440, 1605, 1080,  434,  96,   9;
  9: 16, 224, 1032, 2432, 3405, 3006, 1694, 592, 117, 10;

%F Formula

The row polynomials satisfy (32+(24*n+52)*x+(9*n^2+29*n+20)*x^2+(12*n^2+2*n)*x^3+(4*n^2-4*n)*x^4)*P(n,x) = -(96*n-128+(108*n^2+96*n-344)*x+(27*n^3+195*n^2-128*n-304)*x^2+(54*n^3+91*n^2-201*n-74)*x^3+(36*n^3-12*n^2-74*n+22)*x^4+(8*n^3-12*n^2-4*n+8)*x^5)*P(n-1,x) + (64+(96*n+208)*x+(36*n^2+292*n+232)*x^2+(102*n^2+326*n+80)*x^3+(106*n^2+158*n-28)*x^4+(48*n^2+28*n-24)*x^5+(8*n^2-4)*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[(-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*P[n - 1, x] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*D[P[n - 1, x], x])/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*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] = (-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*v[i - 1] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*deriv(v[i - 1], x))/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*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 shift_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Affine, reversal and Möbius substitutions (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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.