A225117 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 30\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(20 + 352\,x + 4800\,x^{2} - 42\,n - 1476\,n x - 12960\,n x^{2} + 18\,n^{2} + 1080\,n^{2} x + 12393\,n^{2} x^{2} - 216\,n^{3} x - 5022\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-75\,x - 310\,x^{2} + 4825\,x^{3} + 252\,n x + 405\,n x^{2} - 11940\,n x^{3} - 108\,n^{2} x - 540\,n^{2} x^{2} + 10017\,n^{2} x^{3} + 216\,n^{3} x^{2} - 3294\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-63\,x^{2} - 762\,x^{3} + 825\,x^{4} + 189\,n x^{2} + 1701\,n x^{3} - 1890\,n x^{4} - 81\,n^{2} x^{2} - 1296\,n^{2} x^{3} + 1377\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(40 - 500\,x - 39\,n + 900\,n x + 9\,n^{2} - 540\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(96\,x + 200\,x^{2} - 153\,n x - 540\,n x^{2} + 54\,n^{2} x + 432\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	4														6
3	4	52	9													65
4	8	372	540	16												936
5	16	2180	10179	4016	25											16416
6	32	11732	130482	170848	25450	36										338580
7	64	60708	1387017	4199008	2224950	147204	49									8019000
8	128	308404	13281624	78244592	96758800	24929964	802424	64								214326000
9	256	1553284	119437263	1238478928	2943966475	1817136108	253575637	4193728	81							6378341760
10	512	7792020	1032385662	17657593536	72177503850	86002366248	29804754210	2414861568	21232854	100						209118490560
11	1024	39017444	8697386421	234611279232	1534694862000	3125009914920	2117859175866	444828401664	21946218408	104856500	121					7487791113600
12	2048	195214196	72037190340	2967274579152	29550490791600	95114419977960	110412954997800	46191437612160	6200461137168	192699516100	507509332	144				290702478528000
13	4096	976349508	589913712891	36225871623152	529616034330525	2553104687903208	4668023683274910	3369808491972480	921205486528938	82137188894900	1648405199991	2415917232	169			12162363155712000
14	8192	4882353748	4794000440490	430995388031776	8999506364140250	62468956930253436	169811775153266484	193348795292148480	92189964612195108	17166757891379500	1046608952028002	13816726041888	11341395650	196		545481987533683200
15	16384	24413079460	38758459473441	5030719961641376	146869428205752650	1424832364416443292	5521915287497564655	9326511667552051968	7046373391707955788	2318872316868674300	303524588096295599	12939096906675360	113939215448266	52613346436	225	26107021635914419200

Technical details

Node id: n-7ee7d8e0be55305b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9cdd5dbabb33bf1f

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:   2,       4;
   3:   4,      52,          9;
   4:   8,     372,        540,          16;
   5:  16,    2180,      10179,        4016,          25;
   6:  32,   11732,     130482,      170848,       25450,          36;
   7:  64,   60708,    1387017,     4199008,     2224950,      147204,          49;
   8: 128,  308404,   13281624,    78244592,    96758800,    24929964,      802424,         64;
   9: 256, 1553284,  119437263,  1238478928,  2943966475,  1817136108,   253575637,    4193728,       81;
  10: 512, 7792020, 1032385662, 17657593536, 72177503850, 86002366248, 29804754210, 2414861568, 21232854, 100;

%F Formula

The row polynomials satisfy (9*n^2-30*n+25)*P(n,x) = (18*n^2-42*n+20-(216*n^3-1080*n^2+1476*n-352)*x+(729*n^4-5022*n^3+12393*n^2-12960*n+4800)*x^2)*P(n-2,x) - ((108*n^2-252*n+75)*x-(216*n^3-540*n^2+405*n-310)*x^2-(324*n^4-3294*n^3+10017*n^2-11940*n+4825)*x^3)*d/dx P(n-2,x) - ((81*n^2-189*n+63)*x^2-(324*n^3-1296*n^2+1701*n-762)*x^3+(324*n^3-1377*n^2+1890*n-825)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-39*n+40+(108*n^3-540*n^2+900*n-500)*x)*P(n-1,x) + ((54*n^2-153*n+96)*x-(108*n^3-432*n^2+540*n-200)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*P[n - 2, x] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*P[n - 1, x] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 30*n + 25)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*v[i - 2] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*v[i - 1] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 30*n + 25)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.