A253283 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 42\,x + 7\,n + 12\,n x\right)\,P_{n-2}(x) + \left(12 + 18\,x - 7\,n - 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 3\,x\)

\(P_{3}(x) = 5 + 24\,x + 27\,x^{2}\)

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	3														4
3	5	24	27													56
4	25	171	378	270												844
5	129	1152	3780	5400	2835											13296
6	681	7515	32670	69930	73710	30618										215124
7	3653	48024	259875	740880	1173690	979776	336798									3542696
8	19825	302547	1958796	6974100	14747670	18523890	12798324	3752892								59078044
9	108545	1886208	14215824	60691680	160540380	269438400	280215936	165127248	42220035							994444256
10	598417	11665971	100312020	499334220	1585757250	3331942614	4632319692	4109416740	2111001750	478493730						16860842404
11	3317445	71700120	692650035	3938433840	14597012430	36848395584	64165070970	76108649760	58854728790	26795648880	5454828522					287530436376
12	18474633	438439419	4700889270	30057769170	127349872650	375406165134	785690457552	1167524701200	1207239680790	827315659170	338199368364	62482581252				4926424058604
13	103274625	2669784192	31458640356	223417073880	1065108799755	3590967219840	8779445490816	15683876084448	20319236244450	18619148021760	11455139896200	4248815525136	718549684398			84737935739856
14	579168825	16199913483	208079010594	1625031077670	8609182201620	32673572191260	91391621905584	190763838703152	297151783735950	341172249908490	280650927456900	156643831198764	53172676645452	8290957896900		1462370531014644
15	3256957317	98004709656	1362814644555	11607559498080	67652810725500	285429633705216	898979580046548	2147262943082400	3908684651581710	5396134567103280	5561814073801986	4150342977082848	2119721568974100	663276631752000	95938227092700	25308309300757896

Technical details

Node id: n-db2204035897f551 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4f8b2821bd8bee62

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,        3;
   3:      5,       24,        27;
   4:     25,      171,       378,       270;
   5:    129,     1152,      3780,      5400,       2835;
   6:    681,     7515,     32670,     69930,      73710,      30618;
   7:   3653,    48024,    259875,    740880,    1173690,     979776,     336798;
   8:  19825,   302547,   1958796,   6974100,   14747670,   18523890,   12798324,    3752892;
   9: 108545,  1886208,  14215824,  60691680,  160540380,  269438400,  280215936,  165127248,   42220035;
  10: 598417, 11665971, 100312020, 499334220, 1585757250, 3331942614, 4632319692, 4109416740, 2111001750, 478493730;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(12*n-42)*x)*P(n-2,x) - (7*n-12+(12*n-18)*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 3*x, P(3,x) = 5 + 24*x + 27*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[3, x_] := 5 + 24*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (12*n - 42)*x)*P[n - 2, x] - (7*n - 12 + (12*n - 18)*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 + 3*x; v[3] = 5 + 24*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (12*n - 42)*x)*v[i - 2] - (7*n - 12 + (12*n - 18)*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.