A089503 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 5\,x + x^{2} - 6\,n + 3\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(1 - 9\,x + 19\,x^{2} - 10\,x^{3} + 2\,x^{4} - 2\,n + 21\,n x - 43\,n x^{2} + 19\,n x^{3} - 3\,n x^{4} + 2\,n^{2} - 22\,n^{2} x + 35\,n^{2} x^{2} - 12\,n^{2} x^{3} + n^{2} x^{4} + 8\,n^{3} x - 12\,n^{3} x^{2} + 3\,n^{3} x^{3} + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} + 4\,x^{3} - 3\,x^{4} + x^{5} + 2\,n x^{2} - 8\,n x^{3} + 5\,n x^{4} - n x^{5} - 2\,n^{2} x^{2} + 6\,n^{2} x^{3} - 3\,n^{2} x^{4} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	4														9
3	13	72	30													115
4	25	360	1176	336												1897
5	41	1120	9180	23040	5040											38421
6	61	2700	40920	250800	534600	95040										924121
7	85	5544	133770	1485120	7567560	14414400	2162160									25768639
8	113	10192	357840	6232800	56565600	253169280	443963520	57657600								817956945
9	145	17280	831096	20837376	293025600	2303078400	9356256000	15397724160	1764322560							29136092617
10	181	27540	1737360	59179680	1186753680	14299649280	100800806400	379761177600	594255916800	60949324800						1151314573321
11	221	41800	3347190	148579200	4018568400	68485253760	733296564000	4744670515200	16825529952000	25270527744000	2346549004800					49993229570571
12	265	60984	6041640	338635440	11866266720	270988502400	4074253223040	39719171923200	239754131971200	808966322841600	1174305952512000	99638080819200				2366741112797689
13	313	86112	10338900	713856000	31436262000	925256217600	18533257142400	252069801984000	2276177183280000	12971091893760000	41981437802304000	59213488029696000	4626053752320000			121339809137247325
14	365	118300	16923816	1411025616	76243167000	2807997192000	72147819744000	1300522634611200	16289814023683200	137988523632960000	749080556864640000	2340102896651520000	3220427319677568000	233153109116928000		6698417776090081497
15	421	158760	26680290	2642377920	171836584920	7737254965440	247780454922000	5704636025088000	94254144448464000	1101757718565504000	8840046600376992000	46035708446407680000	139496133783726720000	187928263392629760000	12677700308232960000	396179824722628857751

Technical details

Node id: n-51f64a4b5db2f105 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5594e14e270a2e54

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:   5,     4;
   3:  13,    72,      30;
   4:  25,   360,    1176,      336;
   5:  41,  1120,    9180,    23040,       5040;
   6:  61,  2700,   40920,   250800,     534600,       95040;
   7:  85,  5544,  133770,  1485120,    7567560,    14414400,      2162160;
   8: 113, 10192,  357840,  6232800,   56565600,   253169280,    443963520,     57657600;
   9: 145, 17280,  831096, 20837376,  293025600,  2303078400,   9356256000,  15397724160,   1764322560;
  10: 181, 27540, 1737360, 59179680, 1186753680, 14299649280, 100800806400, 379761177600, 594255916800, 60949324800;

%F Formula

The row polynomials satisfy (2*n^2-6*n+5+(3*n-5)*x+x^2)*P(n,x) = (2*n^2-2*n+1+(8*n^3-22*n^2+21*n-9)*x+(2*n^4-12*n^3+35*n^2-43*n+19)*x^2+(3*n^3-12*n^2+19*n-10)*x^3+(n^2-3*n+2)*x^4)*P(n-1,x) - ((2*n^2-2*n+1)*x^2+(2*n^3-6*n^2+8*n-4)*x^3+(3*n^2-5*n+3)*x^4+(n-1)*x^5)*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[Together[((2*n^2 - 2*n + 1 + (8*n^3 - 22*n^2 + 21*n - 9)*x + (2*n^4 - 12*n^3 + 35*n^2 - 43*n + 19)*x^2 + (3*n^3 - 12*n^2 + 19*n - 10)*x^3 + (n^2 - 3*n + 2)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n + 1)*x^2 + (2*n^3 - 6*n^2 + 8*n - 4)*x^3 + (3*n^2 - 5*n + 3)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 5 + (3*n - 5)*x + x^2)]];
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^2 - 2*n + 1 + (8*n^3 - 22*n^2 + 21*n - 9)*x + (2*n^4 - 12*n^3 + 35*n^2 - 43*n + 19)*x^2 + (3*n^3 - 12*n^2 + 19*n - 10)*x^3 + (n^2 - 3*n + 2)*x^4)*v[i - 1] - ((2*n^2 - 2*n + 1)*x^2 + (2*n^3 - 6*n^2 + 8*n - 4)*x^3 + (3*n^2 - 5*n + 3)*x^4 + (n - 1)*x^5)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 5 + (3*n - 5)*x + x^2)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 of numbers used for basis change between certain falling factorials.?
  - 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.