A285061 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(4 + 4\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	41	56	16													113
3	429	796	432	64												1721
4	5329	12336	9568	2816	256											30305
5	75989	208884	207520	90240	16640	1024										600297
6	1215481	3844072	4616176	2630400	730880	92160	4096									13133265
7	21453693	76387980	106951376	74875584	27713280	5325824	487424	16384								313211545
8	412820385	1628917600	2594426816	2144102912	987997696	255696896	35897344	2490368	65536							8062415553
9	8579772325	37066954468	65972453952	62635419904	34438301184	11205949440	2133540864	227868672	12386304	262144						222272909257
10	191166679497	895701311128	1757534758096	1879704773120	1197864993280	469106921472	113076805632	16450191360	1379205120	60293120	1048576					6522046980401
11	4538638641997	22886256582908	48987214377200	58150960067904	42056122380800	19232976345088	5616260407296	1039298101248	119001907200	8030781440	288358400	4194304				202635052145785
12	114238219541617	616028578831760	1426190334050080	1857413136755968	1500441937170176	783839147925504	268903804026880	60569940000768	8849371103232	816804003840	45283803136	1358954496	16777216			6637337932944673
13	3035305413035125	17410732760052948	43293546299049696	61287855655774592	54615716186107648	32051437743463424	12629525228503040	3357603111436288	599111434960896	70697800695808	5364781154304	248529289216	6325010432	67108864		228357151335642281
14	84819458105387417	515186186885014600	1367913300797152944	2088919190322089216	2033110217400627456	1322857413815089152	588474869720657920	180490569510158336	38142656364347392	5509646814543872	535141643452416	33940660289536	1332832370688	29125246976	268435456	8225993954264863377

Technical details

Node id: n-3b2b49a3d24112da · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-3def389a5b57c6c8

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:          5,           4;
  2:         41,          56,          16;
  3:        429,         796,         432,          64;
  4:       5329,       12336,        9568,        2816,         256;
  5:      75989,      208884,      207520,       90240,       16640,        1024;
  6:    1215481,     3844072,     4616176,     2630400,      730880,       92160,       4096;
  7:   21453693,    76387980,   106951376,    74875584,    27713280,     5325824,     487424,     16384;
  8:  412820385,  1628917600,  2594426816,  2144102912,   987997696,   255696896,   35897344,   2490368,    65536;
  9: 8579772325, 37066954468, 65972453952, 62635419904, 34438301184, 11205949440, 2133540864, 227868672, 12386304, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) + (4+4*x)*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[(5 + 4*x)*P[n - 1, x] + (4 + 4*x)*D[P[n - 1, x], 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] = (5 + 4*x)*v[i - 1] + (4 + 4*x)*deriv(v[i - 1], 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).?
  - 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.