A253283 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 56\,x + 7\,n + 16\,n x\right)\,P_{n-2}(x) + \left(12 + 24\,x - 7\,n - 16\,n x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 4\,x\)

\(P_{3}(x) = 5 + 32\,x + 48\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	5	32	48													85
4	25	228	672	640												1565
5	129	1536	6720	12800	8960											30145
6	681	10020	58080	165760	232960	129024										596525
7	3653	64032	462000	1756160	3709440	4128768	1892352									12016405
8	19825	403396	3482304	16531200	46609920	78059520	71909376	28114944								245130485
9	108545	2514944	25272576	143861760	507386880	1135411200	1574436864	1237057536	421724160							5047774465
10	598417	15554628	178332480	1183607040	5011776000	14040778752	26027409408	30785863680	21086208000	6372720640						104702849045
11	3317445	95600160	1231377840	9335546880	46133767680	155278835712	360521441280	570171064320	587883479040	356872355840	96865353728					2184392139925
12	18474633	584585892	8357136480	71248045440	402488486400	1581958490112	4414524162048	8746559078400	12058780631040	11018433986560	6005651931136	1479398129664				45788003137805
13	103274625	3559712256	55926471744	529581212160	3366269786880	15132306309120	49328681385984	117496399527936	202963186483200	247975305543680	203417242828800	100599072817152	22684104654848			963551740008385
14	579168825	21599884644	369918241056	3851925517440	27209267205120	137686164295680	513498056687616	1429115104395264	2968166331187200	4543832661688320	4983722449305600	3708851111067648	1678623744458752	348986225459200		20343935138562365
15	3256957317	130672946208	2422781590320	27514215106560	213816290688000	1202798127218688	5051056735076352	16086308211916800	39042761366568960	71867311891415040	98765173910667264	98267541364801536	66918108731801600	27918898036736000	5384358907084800	430748204500575445

Technical details

Node id: n-73196e5694372c86 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-a7df9941ce8978da

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,        4;
   3:      5,       32,        48;
   4:     25,      228,       672,        640;
   5:    129,     1536,      6720,      12800,       8960;
   6:    681,    10020,     58080,     165760,     232960,      129024;
   7:   3653,    64032,    462000,    1756160,    3709440,     4128768,     1892352;
   8:  19825,   403396,   3482304,   16531200,   46609920,    78059520,    71909376,    28114944;
   9: 108545,  2514944,  25272576,  143861760,  507386880,  1135411200,  1574436864,  1237057536,   421724160;
  10: 598417, 15554628, 178332480, 1183607040, 5011776000, 14040778752, 26027409408, 30785863680, 21086208000, 6372720640;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(16*n-56)*x)*P(n-2,x) - (7*n-12+(16*n-24)*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 5 + 32*x + 48*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 5 + 32*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (16*n - 56)*x)*P[n - 2, x] - (7*n - 12 + (16*n - 24)*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 + 4*x; v[3] = 5 + 32*x + 48*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (16*n - 56)*x)*v[i - 2] - (7*n - 12 + (16*n - 24)*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.