A357613 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 10\,n^{2} + 2\,n^{2} x + 12\,n^{3} - 8\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(-6 - 12\,x - 6\,x^{2} + 45\,n + 82\,n x + 35\,n x^{2} - 108\,n^{2} - 165\,n^{2} x - 54\,n^{2} x^{2} - n^{2} x^{3} + 81\,n^{3} + 63\,n^{3} x - 15\,n^{3} x^{2} - n^{3} x^{3} + 54\,n^{4} x + 48\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-3\,x - 6\,x^{2} - 3\,x^{3} + 18\,n x + 32\,n x^{2} + 14\,n x^{3} - 27\,n^{2} x - 39\,n^{2} x^{2} - 13\,n^{2} x^{3} - n^{2} x^{4} - 18\,n^{3} x^{2} - 20\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	15	40	18													73
3	84	336	315	80												815
4	495	2640	3780	2016	350											9281
5	3003	20020	38610	31680	11550	1512										106375
6	18564	148512	360360	400400	225225	61776	6468									1221305
7	116280	1085280	3174444	4455360	3403400	1441440	315315	27456								14018975
8	735471	7845024	26860680	45581760	44089500	25395552	8576568	1555840	115830							160756225
9	4686825	56241900	220641300	439321344	514829700	376049520	172830840	48372480	7482618	486200						1840942727
10	30045015	400600200	1771619850	4049416800	5577321750	4942365120	2883046320	1098303360	261891630	35271600	2032316					21051913961
11	193536720	2838538560	13970931975	36054018000	57085528500	59526426960	42164552430	20397062400	6619239000	1372879200	163601438	8465184				240394780367
12	1251677700	20026843200	108574099920	312239241600	558837279000	670604734800	559438179300	328812644160	135528918525	38244492000	7011490200	748843200	35154028			2741353597633
13	8122425444	140788707696	833617348200	2643543302400	5277907635000	7165890594720	6884875277280	4768744780800	2389029367725	856282927500	213499876590	35045861760	3389852700	145608400		31220883566215
14	52860229080	986724276160	6335491846320	21963038400576	48396118270500	73358326641600	79801963441200	63696805286400	37620925622280	16392560184000	5190977391600	1158317269200	172035024525	15201516960	601749000	355141947149401

Technical details

Node id: n-7fb3d7e99783adbd · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2dbb8918a8b74f8e

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:       3,        4;
  2:      15,       40,        18;
  3:      84,      336,       315,        80;
  4:     495,     2640,      3780,      2016,       350;
  5:    3003,    20020,     38610,     31680,     11550,      1512;
  6:   18564,   148512,    360360,    400400,    225225,     61776,      6468;
  7:  116280,  1085280,   3174444,   4455360,   3403400,   1441440,    315315,    27456;
  8:  735471,  7845024,  26860680,  45581760,  44089500,  25395552,   8576568,  1555840,  115830;
  9: 4686825, 56241900, 220641300, 439321344, 514829700, 376049520, 172830840, 48372480, 7482618, 486200;

%F Formula

The row polynomials satisfy (12*n^3-10*n^2+2*n+(8*n^4-8*n^3+2*n^2)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+63*n^3-165*n^2+82*n-12)*x+(48*n^4-15*n^3-54*n^2+35*n-6)*x^2+(2*n^4-n^3-n^2)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+39*n^2-32*n+6)*x^2+(20*n^3+13*n^2-14*n+3)*x^3+(2*n^3+n^2)*x^4)*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[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 2*n^2)*x)]];
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] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 2*n^2)*x)); 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 Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).?
  - 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

References

[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.