A257614 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-13 + 10\,x + 5\,n\right)\,P_{n}(x) = \left(112 - 636\,x + 1852\,x^{2} - 1860\,x^{3} + 400\,x^{4} - 70\,n + 470\,n x - 2010\,n x^{2} + 1700\,n x^{3} - 250\,n x^{4} - 100\,n^{2} x + 850\,n^{2} x^{2} - 375\,n^{2} x^{3} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(80\,x - 700\,x^{2} + 1470\,x^{3} - 1100\,x^{4} + 250\,x^{5} - 50\,n x + 575\,n x^{2} - 900\,n x^{3} + 375\,n x^{4} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-107 + 363\,x - 300\,x^{2} + 50\,x^{3} + 45\,n - 235\,n x + 125\,n x^{2} + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-65\,x + 115\,x^{2} - 50\,x^{3} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	32	36	4													72
3	252	732	268	8												1260
4	1860	13556	11388	1908	16											28728
5	13292	216500	406480	154660	13420	32										804384
6	93748	3116004	11876400	9499720	1979580	94068	64									26659584
7	657964	41984612	296609028	459909400	195298820	24610404	658732	128								1019729088
8	4609844	542632404	6633821716	18310414316	14655219900	3735671884	301278524	4611636	256							44188260480
9	32278380	6829661172	137425562056	629962633916	877930515672	414375493420	68466465032	3656914596	32282476	512						2138711807232
10	225970164	84505909700	2697851540520	19480031223840	44155571713680	35793565691688	10845358839000	1222356952320	44173692540	225979380	1024					114323867513856
11	1581839276	1034182223108	50973033142300	556919483857680	1945502481110520	2534668273048632	1307271470108856	269576431083600	21465612719580	532118565220	1581859756	2048				6687946249560576
12	11072981428	12566788679604	936840645194796	15017977244185700	77539487030630280	153817167473802216	126581665390956888	44188087220514792	6468635774132100	372966174292580	6399660601788	11073026484	4096			424941815549002752
13	77511103468	152008418938388	16874878663540992	387486278008307732	2861919186309464500	8282215369914083496	10305234486221038656	5714470352764065864	1412534445859782324	151468037433340100	6435257729676928	76895587064868	77511201772	8192		29138867351931617280
14	542578232180	1833324848147364	299488008319082272	9665754607573880632	99521481008282267124	406180567656775157420	731723030148125196768	610996196320048157904	239292959149390395804	43341561502845432508	3486320280136554240	110520566014294904	923444651771132	542578445172	16384	2144620637102167031808

Technical details

Node id: n-358c98927e7dd542 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-103989f75d6354bd

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        4,          2;
  2:       32,         36,            4;
  3:      252,        732,          268,            8;
  4:     1860,      13556,        11388,         1908,           16;
  5:    13292,     216500,       406480,       154660,        13420,           32;
  6:    93748,    3116004,     11876400,      9499720,      1979580,        94068,          64;
  7:   657964,   41984612,    296609028,    459909400,    195298820,     24610404,      658732,        128;
  8:  4609844,  542632404,   6633821716,  18310414316,  14655219900,   3735671884,   301278524,    4611636,      256;
  9: 32278380, 6829661172, 137425562056, 629962633916, 877930515672, 414375493420, 68466465032, 3656914596, 32282476, 512;

%F Formula

The row polynomials satisfy (5*n-13+10*x)*P(n,x) = -(70*n-112+(100*n^2-470*n+636)*x+(125*n^3-850*n^2+2010*n-1852)*x^2+(375*n^2-1700*n+1860)*x^3+(250*n-400)*x^4)*P(n-2,x) - ((50*n-80)*x+(125*n^2-575*n+700)*x^2-(125*n^2-900*n+1470)*x^3-(375*n-1100)*x^4-250*x^5)*d/dx P(n-2,x) + (45*n-107+(50*n^2-235*n+363)*x+(125*n-300)*x^2+50*x^3)*P(n-1,x) + ((25*n-65)*x-(25*n-115)*x^2-50*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n - 112 + (100*n^2 - 470*n + 636)*x + (125*n^3 - 850*n^2 + 2010*n - 1852)*x^2 + (375*n^2 - 1700*n + 1860)*x^3 + (250*n - 400)*x^4)*P[n - 2, x] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 13 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n - 112 + (100*n^2 - 470*n + 636)*x + (125*n^3 - 850*n^2 + 2010*n - 1852)*x^2 + (375*n^2 - 1700*n + 1860)*x^3 + (250*n - 400)*x^4)*v[i - 2] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 13 + 10*x)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.