A269944 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	1	2													3
3	0	4	13	10												27
4	0	36	157	220	100											513
5	0	576	3124	6189	5340	1700										16929
6	0	14400	93076	235949	294414	181340	44200									863379
7	0	518400	3883536	11937976	19329017	17421558	8300780	1635400								63026667
8	0	25401600	216213264	779137624	1544020633	1820107192	1277816120	495173600	81770000							6239640033
9	0	1625702400	15488752896	63918670096	149461266072	216848201433	200087199160	114749158200	37419564000	5315050000						804913564257
10	0	131681894400	1387896581376	6447490015248	17347693499704	29820528133977	33988615649466	25701832145320	12440415656400	3498923298000	435834100000					131200910973891
11	0	13168189440000	152089529472000	784926556243776	2385965841510448	4734169856867804	6410734906478277	6003033395128066	3839926612317320	1606374311096400	396974663098000	44019244100000				26371383105752091
12	0	1593350922240000	20009352177792000	113531035901080896	384462906684504880	863922385345278940	1353267646221743605	1508476699400845780	1197001194296019772	662842338345377440	244011600188618800	53757237434056000	5370347780200000			6408246094697758113
13	0	229442532802560000	3112382597326848000	19249825235535489024	71824658768225432640	180151944958973374960	320139286930996525420	413444453415874615045	391097293391749485268	269014469894657218300	131249809487240836000	43122724217853790000	8568129508286920000	778700428129000000		1851983121367652094657
14	0	38775788043632640000	564997889524672512000	3782325506351061805056	15410837621871131250240	42655870688664823917040	84729370134363886539180	124295791405552219264005	136380999663904347567942	111949985288794482388260	67913677685435428395000	29600208005648232630000	8778877003935633780000	1588182388762577400000	132379072781930000000	627822278143634060088723

Technical details

Node id: n-3e9ab13cc75c9520 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-ce583c2d05188b3b

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: 1;
  1: 0,          1;
  2: 0,          1,           2;
  3: 0,          4,          13,          10;
  4: 0,         36,         157,         220,          100;
  5: 0,        576,        3124,        6189,         5340,         1700;
  6: 0,      14400,       93076,      235949,       294414,       181340,        44200;
  7: 0,     518400,     3883536,    11937976,     19329017,     17421558,      8300780,      1635400;
  8: 0,   25401600,   216213264,   779137624,   1544020633,   1820107192,   1277816120,    495173600,    81770000;
  9: 0, 1625702400, 15488752896, 63918670096, 149461266072, 216848201433, 200087199160, 114749158200, 37419564000, 5315050000;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n^2-2*n+1+(2*n^2-4*n+3)*x+(n^2-2*n+2)*x^2)*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[Together[((n^2 - 2*n + 1 + (2*n^2 - 4*n + 3)*x + (n^2 - 2*n + 2)*x^2)*P[n - 1, x])/(1 + 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] = ((n^2 - 2*n + 1 + (2*n^2 - 4*n + 3)*x + (n^2 - 2*n + 2)*x^2)*v[i - 1])/(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, 0]
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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
  - 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
  - Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.