A029635 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2 - 4\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	8														11
2	6	28	32													66
3	12	80	176	128												396
4	24	208	672	960	512											2376
5	48	512	2176	4608	4864	2048										14256
6	96	1216	6400	17920	28160	23552	8192									85536
7	192	2816	17664	61440	128000	159744	110592	32768								513216
8	384	6400	46592	193536	501760	831488	860160	507904	131072							3079296
9	768	14336	118784	573440	1777664	3670016	5046272	4456448	2293760	524288						18475776
10	1536	31744	294912	1622016	5849088	14450688	24772608	29097984	22413312	10223616	2097152					110854656
11	3072	69632	716800	4423680	18186240	52297728	107347968	157286400	161218560	110100480	45088768	8388608				665127936
12	6144	151552	1712128	11714560	54067200	177340416	423886848	743964672	951582720	865075200	530579456	197132288	33554432			3990767616
13	12288	327680	4030464	30277632	154992640	570949632	1557135360	3183476736	4879024128	5536481280	4521459712	2516582400	855638016	134217728		23944605696
14	24576	704512	9371648	76677120	431095808	1761869824	5398069248	12595494912	22491955200	30589059072	31188844544	23119003648	11777605632	3690987520	536870912	143667634176

Technical details

Node id: n-e32b221435f26eba · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-f052328eca4fdea3

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   2;
  1:   3,     8;
  2:   6,    28,     32;
  3:  12,    80,    176,    128;
  4:  24,   208,    672,    960,     512;
  5:  48,   512,   2176,   4608,    4864,    2048;
  6:  96,  1216,   6400,  17920,   28160,   23552,    8192;
  7: 192,  2816,  17664,  61440,  128000,  159744,  110592,   32768;
  8: 384,  6400,  46592, 193536,  501760,  831488,  860160,  507904,  131072;
  9: 768, 14336, 118784, 573440, 1777664, 3670016, 5046272, 4456448, 2293760, 524288;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (4*n - 4)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (4*n - 4)*x)*v[i - 1])/(n - 1)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from The (1,2)-Pascal triangle (or Lucas triangle) read by rows.?
  - 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
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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
[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.