A253284 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	24	42	12													78
3	168	624	600	120												1512
4	1320	8400	16200	10920	1680											38520
5	11520	111600	352800	463680	241920	30240										1211760
6	110880	1517040	7056000	14817600	14817600	6320160	665280									45304560
7	1169280	21450240	137168640	414892800	651974400	526901760	190270080	17297280								1961124480
8	13426560	317882880	2667168000	10897286400	24309331200	30512401920	20704844160	6486480000	518918400							96427739520
9	166924800	4953312000	52690176000	278299929600	828193766400	1460236377600	1525620096000	892539648000	247005158400	17643225600						5307348614400
10	2235340800	81230688000	1066976064000	7055214566400	26850913689600	62572218508800	91101314304000	81511702272000	41920304025600	10391859878400	670442572800					323224411910400
11	32093107200	1401558681600	22261599360000	179805225600000	848680664832000	2508642507571200	4792888093593600	5938709736960000	4646167029504000	2132007381504000	478695996979200	28158588057600				21577450475750400
12	491934643200	25418698905600	479960082201600	4643769626496000	26553635716608000	96640274213683200	232699607646105600	376206781760409600	405544007860992000	281867466456576000	116773664674867200	23962958437017600	1295295050649600			1566693332159155200
13	8020402790400	483926694451200	10709853073920000	122171657287680000	830767269556224000	3639269273678438400	10727070437718835200	21744061698078720000	30441686377310208000	29032349045027328000	18152264839803494400	6854701408037683200	1295295050649600000	64764752532480000		122915603609851852800
14	138613483008000	9657175207680000	247516604375040000	3285306929608704000	26169168991021056000	135537984584690688000	478369357357731840000	1181427352262277120000	2064961059260875776000	2547800029300990464000	2178271780776419328000	1237654420895692800000	429390309290342400000	75191877690209280000	3497296636753920000	10361813256369679104000

Technical details

Node id: n-854b531da9a10af4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-695598672c6e8bd2

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:         4,          2;
  2:        24,         42,          12;
  3:       168,        624,         600,          120;
  4:      1320,       8400,       16200,        10920,         1680;
  5:     11520,     111600,      352800,       463680,       241920,         30240;
  6:    110880,    1517040,     7056000,     14817600,     14817600,       6320160,        665280;
  7:   1169280,   21450240,   137168640,    414892800,    651974400,     526901760,     190270080,     17297280;
  8:  13426560,  317882880,  2667168000,  10897286400,  24309331200,   30512401920,   20704844160,   6486480000,    518918400;
  9: 166924800, 4953312000, 52690176000, 278299929600, 828193766400, 1460236377600, 1525620096000, 892539648000, 247005158400, 17643225600;

%F Formula

The row polynomials satisfy (n^2-n+2+x-x^2)*P(n,x) = (n^3+2*n^2+3*n+2+(2*n^3+2*n^2+3*n+1)*x-(n+1)*x^2-2*n*x^3)*P(n-1,x) + ((2*n^2+2)*x+2*n^2*x^2-4*x^3-2*x^4)*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[((n^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*P[n - 1, x] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^2 - n + 2 + x - x^2)]];
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^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*v[i - 1] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^2 - n + 2 + x - x^2)); 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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

Related ?

Same fingerprint ?

A088617 → D+1

References

[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.