A253283 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 28\,x + 7\,n + 8\,n x\right)\,P_{n-2}(x) + \left(12 + 12\,x - 7\,n - 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 2\,x\)

\(P_{3}(x) = 5 + 16\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	5	16	12						33
4	25	114	168	80					387
5	129	768	1680	1600	560				4737
6	681	5010	14520	20720	14560	4032			59523
7	3653	32016	115500	219520	231840	129024	29568		761121
8	19825	201698	870576	2066400	2913120	2439360	1123584	219648	9854211

All 15 rows

Technical details

Node id: n-57fab0d047111899 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c15c2b3e5c2d6a7d

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:      1,       2;
   3:      5,      16,       12;
   4:     25,     114,      168,        80;
   5:    129,     768,     1680,      1600,       560;
   6:    681,    5010,    14520,     20720,     14560,      4032;
   7:   3653,   32016,   115500,    219520,    231840,    129024,     29568;
   8:  19825,  201698,   870576,   2066400,   2913120,   2439360,   1123584,    219648;
   9: 108545, 1257472,  6318144,  17982720,  31711680,  35481600,  24600576,   9664512,  1647360;
  10: 598417, 7777314, 44583120, 147950880, 313236000, 438774336, 406678272, 240514560, 82368000, 12446720;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(8*n-28)*x)*P(n-2,x) - (7*n-12+(8*n-12)*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 5 + 16*x + 12*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 5 + 16*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (8*n - 28)*x)*P[n - 2, x] - (7*n - 12 + (8*n - 12)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 5 + 16*x + 12*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (8*n - 28)*x)*v[i - 2] - (7*n - 12 + (8*n - 12)*x)*v[i - 1])/(-n + 1)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/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.