A109983 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} - n - 6\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 15\,x - 18\,x^{2} + 6\,n + 30\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 15\,x + 18\,x^{2}\)

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	15	18													36
2	13	132	495	810	486											1936
3	63	963	6075	20223	37422	36450	14580									115776
4	321	6552	58104	292248	911331	1803060	2208870	1530900	459270							7270656
5	1683	42975	491130	3306960	14523705	43456905	89681580	125993070	115276770	62001450	14880348					469656576
6	8989	275580	3854979	32529330	184372605	739293480	2149904835	4567440150	7033720050	7655112360	5587570674	2455257420	491051484			30900391936
7	48639	1740291	28800135	292143915	2029014603	10204970895	38323555515	109146444975	236853346590	389654312670	478286625474	424677691746	257802029100	95755039380	16415149608	2059470913536

All 15 rows

Technical details

Node id: n-9906787568ecda50 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-c986abc9710a95e7

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,       15,         18;
  2:      13,      132,        495,         810,          486;
  3:      63,      963,       6075,       20223,        37422,         36450,         14580;
  4:     321,     6552,      58104,      292248,       911331,       1803060,       2208870,        1530900,          459270;
  5:    1683,    42975,     491130,     3306960,     14523705,      43456905,      89681580,      125993070,       115276770,        62001450,        14880348;
  6:    8989,   275580,    3854979,    32529330,    184372605,     739293480,    2149904835,     4567440150,      7033720050,      7655112360,      5587570674,       2455257420,        491051484;
  7:   48639,  1740291,   28800135,   292143915,   2029014603,   10204970895,   38323555515,   109146444975,    236853346590,    389654312670,    478286625474,     424677691746,     257802029100,      95755039380,      16415149608;
  8:  265729, 10868784,  207697320,  2461176144,  20239296336,  122457847680,  563849657280,  2015111072880,   5648399771715,  12457404554760,  21542721249924,   28899689304024,   29479048213230,   22100263088904,   11482397150796,    3693408661800,    554011299270;
  9: 1462563, 67312431, 1458664362, 19773754344, 187892308368, 1328824874160, 7248199140000, 31175218731600, 107174061660105, 296660980356165, 662682681598698, 1191648175213716, 1712004594600762, 1938206478602394, 1690940976269688, 1096819258932540, 498056158043730, 141272881313850, 18836384175180;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-1+(6*n-6)*x+(9*n-9)*x^2)*P(n-2,x) + (6*n-3+(30*n-15)*x+(36*n-18)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 15*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*P[n - 2, x] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*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 + 15*x + 18*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*v[i - 2] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*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.