A161198 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	4														11
3	63	64	16													143
4	693	956	432	64												2145
5	9009	15200	9440	2560	256											36465
6	135135	264036	202400	76160	14080	1024										692835
7	2297295	5029152	4496944	2104320	544000	73728	4096									14549535
8	43648605	104743068	105558544	57969856	18753280	3576832	372736	16384								334639305
9	916620705	2374198848	2635701696	1639601152	625698304	150126592	22134784	1835008	65536							8365982625
10	21082276215	58273056324	70117934400	48253633280	20949465600	5955704832	1109606400	130744320	8847360	262144						225881530875
11	527056905375	1541155512960	1986040585296	1486812569600	716751173120	232690483200	51562979328	7707033600	744161280	41943040	1048576					6550564395375
12	14230536445125	43719426471420	59787717854832	48088101720384	25299531952640	9149647738880	2322962374656	414341824512	50920488960	4109107200	196083712	4194304				203067496256625
13	412685556908625	1324785513451680	1908721523675808	1633705821310464	926038833508096	366537912238080	103964499820544	21307762409472	3134061477888	322846064640	22122856448	905969664	16777216			6701227376468625
14	12793252264167375	42719093144636580	64469509287756768	58279766555327616	35242027123992832	15066830613412864	4689051143389184	1076398633975808	182386955452416	22544473915392	1977192808448	116576485376	4143972352	67108864		234542958176401875
15	422177324717523375	1460903082829676640	2298370179074519664	2181110333476838400	1396105961313073920	638173518738595840	215006010185494528	54277359494758400	10324364065832960	1473515461017600	155425258340352	11755795251200	603057029120	18790481920	268435456	8678089452526869375

Technical details

Node id: n-9e117970c0630b95 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-fef1c3c767bad281

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           7,           4;
   3:          63,          64,          16;
   4:         693,         956,         432,          64;
   5:        9009,       15200,        9440,        2560,         256;
   6:      135135,      264036,      202400,       76160,       14080,       1024;
   7:     2297295,     5029152,     4496944,     2104320,      544000,      73728,       4096;
   8:    43648605,   104743068,   105558544,    57969856,    18753280,    3576832,     372736,     16384;
   9:   916620705,  2374198848,  2635701696,  1639601152,   625698304,  150126592,   22134784,   1835008,   65536;
  10: 21082276215, 58273056324, 70117934400, 48253633280, 20949465600, 5955704832, 1109606400, 130744320, 8847360, 262144;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 3 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n + 3 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

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.