A225467 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 - 5\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	16	40	9													65
4	64	336	316	27												743
5	256	2304	4960	2320	81											9921
6	1024	14080	54400	63840	16564	243										150151
7	4096	79872	485120	1071360	768496	116920	729									2526593
8	16384	430080	3777536	13502720	19144384	8921136	821356	2187								46615783
9	65536	2228224	26722304	140894208	333129216	322850304	101417920	5758240	6561							933072513
10	262144	11206656	175964160	1285079040	4573083648	7620856320	5248426240	1138630080	40333924	19683						20093861895
11	1048576	55050240	1096089600	10595205120	52989468672	135664349184	165789975040	83280913920	12686266576	282416200	59049					462440842177
12	4194304	265289728	6531317760	80743956480	540409233408	1973373050880	3783439931392	3483133181440	1299958775104	140678597136	1977149596	177147				11310514854375
13	16777216	1258291200	37532729344	577697218560	4987675410432	24646178439168	68414832099328	100951651147776	71379365547776	20062096015104	1555373166880	13840755760	531441			292627518129985
14	67108864	5888802816	209270603776	3924696236032	42480893165568	273153353121792	1037690860011520	2251007071272960	2607405438589952	1436456329468160	307152932894080	17164467858720	96887416084	1594323		7976748158144647
15	268435456	27246198784	1137411358720	25534503321600	338685510811648	2749368245944320	13711130799308800	41172444945448960	71206812678729728	65716150405441536	28521281991471360	4675951864846080	189196696110256	678218289880	4782969	228308400790500097

Technical details

Node id: n-9347079e8aa5e773 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2498353059445bbe

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      4,        3;
   3:     16,       40,         9;
   4:     64,      336,       316,         27;
   5:    256,     2304,      4960,       2320,         81;
   6:   1024,    14080,     54400,      63840,      16564,        243;
   7:   4096,    79872,    485120,    1071360,     768496,     116920,        729;
   8:  16384,   430080,   3777536,   13502720,   19144384,    8921136,     821356,       2187;
   9:  65536,  2228224,  26722304,  140894208,  333129216,  322850304,  101417920,    5758240,     6561;
  10: 262144, 11206656, 175964160, 1285079040, 4573083648, 7620856320, 5248426240, 1138630080, 40333924, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4+(4*n-5)*x)*P(n-1,x) - 4*x^2*d/dx 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[(4 + (4*n - 5)*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], 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] = (4 + (4*n - 5)*x)*v[i - 1] - 4*x^2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - 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

Related ?

Same fingerprint ?

A225467: Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of or…

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.