A137320 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + x + n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	2														8
2	36	30	6													72
3	216	324	156	24												720
4	1296	3024	2556	924	120											7920
5	7776	25920	33480	20880	6264	720										95040
6	46656	209952	382320	359640	183744	48168	5040									1235520
7	279936	1632960	3973536	5216400	3979584	1758960	415584	40320								17297280
8	1679616	12317184	38537856	67060224	70825104	46370016	18324144	3982176	362880							259459200
9	10077696	90699264	354398976	787739904	1095552864	986471136	573645024	207134496	41999040	3628800						4151347200
10	60466176	655050240	3124085760	8624828160	15238456128	17969908320	14293052640	7552902240	2530473696	483762240	39916800					70572902400
11	362797056	4655895552	26605117440	89237998080	194928674688	290680923456	301397215680	216834045120	105817669056	33268257792	6044647680	479001600				1270312243200
12	2176782336	32651735040	220157346816	881294515200	2329666023168	4278158311680	5587235299008	5219168074560	3453748600896	1575239244480	468755237376	81454429440	6227020800			24135932620800
13	13060694016	226385362944	1778068371456	8369969946624	26316119351808	58284274194432	93417628157568	109536302633472	93790844649216	57803915879424	24865880846976	7051299899904	1177724136960	87178291200		482718652416000
14	78364164096	1554222587904	14064190672896	76890845251584	283446265310208	744447435443712	1434769881861888	2058482238164352	2205789607397376	1753686165014784	1016254023273216	415296012104064	112835843320320	18188931801600	1307674368000	10137091700736000

Technical details

Node id: n-28e86633924230ef · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-fe4d40ee62c150a8

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:        6,        2;
  2:       36,       30,         6;
  3:      216,      324,       156,        24;
  4:     1296,     3024,      2556,       924,        120;
  5:     7776,    25920,     33480,     20880,       6264,       720;
  6:    46656,   209952,    382320,    359640,     183744,     48168,      5040;
  7:   279936,  1632960,   3973536,   5216400,    3979584,   1758960,    415584,     40320;
  8:  1679616, 12317184,  38537856,  67060224,   70825104,  46370016,  18324144,   3982176,   362880;
  9: 10077696, 90699264, 354398976, 787739904, 1095552864, 986471136, 573645024, 207134496, 41999040, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (6+(n+1)*x)*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[(6 + (n + 1)*x)*P[n - 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] = (6 + (n + 1)*x)*v[i - 1]); 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 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 basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A137320 → basis A013610

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.