A109983 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 8\,x + 16\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 20\,x - 32\,x^{2} + 6\,n + 40\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 20\,x + 32\,x^{2}\)

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	20	32													55
2	13	176	880	1920	1536											4525
3	63	1284	10800	47936	118272	153600	81920									413875
4	321	8736	103296	692736	2880256	7598080	12410880	11468800	4587520							39750625
5	1683	57300	873120	7838720	45902080	183127040	503889920	943882240	1151467520	825753600	264241152					3927034375
6	8989	367440	6853296	77106560	582708480	3115376640	12079575040	34217164800	70257868800	101953044480	99222552576	58133053440	15502147584			395147828125
7	48639	2320388	51200240	692489280	6412688128	43003663360	215326863360	817675059200	2365862051840	5189531074560	8493283147776	10055080476672	8138627481600	4030558371840	921270484992	40277377421875

All 15 rows

Technical details

Node id: n-da7295b3362987cc · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-487455b1f16acaff

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,       20,         32;
  2:      13,      176,        880,        1920,         1536;
  3:      63,     1284,      10800,       47936,       118272,        153600,          81920;
  4:     321,     8736,     103296,      692736,      2880256,       7598080,       12410880,        11468800,          4587520;
  5:    1683,    57300,     873120,     7838720,     45902080,     183127040,      503889920,       943882240,       1151467520,        825753600,         264241152;
  6:    8989,   367440,    6853296,    77106560,    582708480,    3115376640,    12079575040,     34217164800,      70257868800,     101953044480,       99222552576,       58133053440,       15502147584;
  7:   48639,  2320388,   51200240,   692489280,   6412688128,   43003663360,   215326863360,    817675059200,    2365862051840,    5189531074560,     8493283147776,    10055080476672,     8138627481600,     4030558371840,      921270484992;
  8:  265729, 14491712,  369239680,  5833899008,  63966171136,  516036362240,  3168077086720,  15096287068160,   56420290723840,  165911388487680,   382549754904576,   684257043283968,   930632674836480,   930252872220672,   644428704251904,   276381145497600,    55276229099520;
  9: 1462563, 89749908, 2593181088, 46871121408, 593832480768, 5599657082880, 40725135360000, 233550426931200, 1070531825172480, 3951018444062720, 11767737904791552, 28214616718835712, 54046772598669312, 81583741171924992, 94900993929510912, 82075987507937280, 49693329960468480, 18793917893836800, 3341140958904320;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-1+(8*n-8)*x+(16*n-16)*x^2)*P(n-2,x) + (6*n-3+(40*n-20)*x+(64*n-32)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 20*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*P[n - 2, x] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*x^2)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 20*x + 32*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*v[i - 2] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*x^2)*v[i - 1])/(n)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.