A357613 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 6\,x - 54\,n - 28\,n x - n x^{2} + 52\,n^{2} + 26\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(6 + 9\,x + 3\,x^{2} - 20\,n - 32\,n x - 13\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)

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

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	2														7
2	41	32	6													79
3	377	438	165	20												1000
4	3649	5632	3192	784	70											13327
5	36365	70010	53010	19680	3570	252										182887
6	369305	852000	807510	401720	110385	15840	924									2557684
7	3800305	10218670	11633895	7259980	2677675	582582	69069	3432								36245608
8	39490049	121266176	161196000	121024960	56066920	16388736	2946944	297440	12870							518690095
9	413442773	1427499954	2169942084	1904517888	1062646200	390478032	94379376	14448096	1268982	48620						7478672005
10	4354393801	16697520160	28568732310	28701839280	18737291370	8298484992	2522733360	519228960	69157530	5374720	184756					108474941239
11	46082942185	194311838182	369549368265	418219637220	312749562510	162162928620	59445520134	15393301440	2756737170	324815260	22632610	705432				1581019989028
12	489658242241	2251695355392	4712199116856	5931679581520	4999738028670	2971189755072	1275714885984	398475765072	89797331745	14225643520	1502322096	94853472	2704156			23135973585796
13	5220495115997	26000388450218	59375640364950	82289853245600	77190403540250	51728087911500	25465477837800	9319511744400	2533852260225	505149123050	71719045970	6860830080	395965700	10400600		339707845836340
14	55818956905529	299322990243872	740694270818502	1120726146280104	1157963637305475	863938660384800	479782524721500	201380564127600	64164574348875	15435319099200	2757808443390	354624608520	30999768345	1647455040	40116600	5002372764627352

Technical details

Node id: n-f721652c141580b8 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-52349188d9f9d3f2

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,          2;
  2:        41,         32,          6;
  3:       377,        438,        165,         20;
  4:      3649,       5632,       3192,        784,         70;
  5:     36365,      70010,      53010,      19680,       3570,       252;
  6:    369305,     852000,     807510,     401720,     110385,     15840,      924;
  7:   3800305,   10218670,   11633895,    7259980,    2677675,    582582,    69069,     3432;
  8:  39490049,  121266176,  161196000,  121024960,   56066920,  16388736,  2946944,   297440,   12870;
  9: 413442773, 1427499954, 2169942084, 1904517888, 1062646200, 390478032, 94379376, 14448096, 1268982, 48620;

%F Formula

The row polynomials satisfy (4*n^2-2*n)*P(n,x) = (52*n^2-54*n+12+(26*n^2-28*n+6)*x+(n^2-n)*x^2)*P(n-1,x) - (20*n-6+(32*n-9)*x+(13*n-3)*x^2+n*x^3)*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[Together[((52*n^2 - 54*n + 12 + (26*n^2 - 28*n + 6)*x + (n^2 - n)*x^2)*P[n - 1, x] - (20*n - 6 + (32*n - 9)*x + (13*n - 3)*x^2 + n*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n)]];
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] = ((52*n^2 - 54*n + 12 + (26*n^2 - 28*n + 6)*x + (n^2 - n)*x^2)*v[i - 1] - (20*n - 6 + (32*n - 9)*x + (13*n - 3)*x^2 + n*x^3)*deriv(v[i - 1], x))/(4*n^2 - 2*n)); 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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).?
  - 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

References

[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.