A144439 → D+1

nonnegative coefficients

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

Recurrence

\(\left(16\right)\,P_{n}(x) = \left(52 - 70\,x - 112\,x^{2} - 108\,x^{3} - 128\,n - 64\,n x + 176\,n x^{3} + 48\,n^{2} + 24\,n^{2} x - 48\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(26\,x - 48\,x^{2} - 108\,x^{3} - 64\,n x + 176\,n x^{3} + 24\,n^{2} x - 48\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-169 + 475\,x + 418\,x^{2} + 142\,x^{3} + 468\,n - 134\,n x - 88\,n x^{2} - 548\,n x^{3} - 284\,n^{2} - 76\,n^{2} x - 24\,n^{2} x^{2} + 296\,n^{2} x^{3} + 48\,n^{3} + 24\,n^{3} x - 48\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(65 - 263\,x + 184\,x^{2} + 426\,x^{3} + 108\,x^{4} - 160\,n + 378\,n x + 104\,n x^{2} - 724\,n x^{3} - 176\,n x^{4} + 60\,n^{2} - 196\,n^{2} x - 48\,n^{2} x^{2} + 344\,n^{2} x^{3} + 48\,n^{2} x^{4} + 24\,n^{3} x - 48\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(26\,x - 74\,x^{2} - 60\,x^{3} + 108\,x^{4} - 64\,n x + 64\,n x^{2} + 176\,n x^{3} - 176\,n x^{4} + 24\,n^{2} x - 24\,n^{2} x^{2} - 48\,n^{2} x^{3} + 48\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(23 - 200\,x - 298\,x^{2} + 192\,x^{3} - 40\,n + 464\,n x + 200\,n x^{2} - 128\,n x^{3} - 12\,n^{2} - 96\,n^{2} x - 40\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-13 + 208\,x - 226\,x^{2} - 32\,x^{3} + 128\,x^{4} + 32\,n - 48\,n x - 40\,n x^{2} - 12\,n^{2} + 24\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(64\,x^{2} - 128\,x^{3} + 64\,x^{4}\right)\,P_{n-2}''(x) + \left(36 - 208\,x + 64\,x^{2} + 24\,n + 64\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 9 + 10\,x + x^{2}\)

\(P_{3}(x) = 32 + 93\,x + 34\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	9	10	1													20
3	32	93	34	1												160
4	103	754	632	106	1											1596
5	318	5129	9624	3674	322	1										19068
6	965	31138	118095	94684	19907	970	1									265760
7	2908	176253	1229514	1904735	814312	104115	2914	1								4234752
8	8739	954906	11423702	31353078	25176748	6488710	534250	8746	1							75948880
9	26234	5030841	98126944	444857368	619473732	294494902	49391008	2712168	26242	1						1514139440
10	78721	26028634	797789621	5665142688	12777160050	10390561820	3182118666	365404544	13683437	78730	1					33218046912
11	236184	133082045	6236862210	66639945613	230674760200	300340567898	155926277324	32600287874	2654619472	68784761	236194	1				795275659776
12	708575	675172450	47394263700	738889999362	3765062407547	7437579411428	6138087048832	2163260909620	321972175813	19057996634	345026252	708586	1			20632325828800
13	2125750	3407984265	352746301272	7834419332358	56885775492070	163392311184303	203201368439520	113330449892244	28355445823458	3099495792135	135746679176	1728438246	2125762	1		576592899610560
14	6377277	17144791602	2585398321691	80270286747780	809469879520861	3268850033364486	5868205388985755	4912147540224600	1939798449242527	356451767992702	29301209347337	961773524132	8652111751	6377290	1	17268067536929792

Technical details

Node id: n-e77f329a666005a4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-935341af6a0597f9

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:     2,       1;
  2:     9,      10,        1;
  3:    32,      93,       34,         1;
  4:   103,     754,      632,       106,         1;
  5:   318,    5129,     9624,      3674,       322,         1;
  6:   965,   31138,   118095,     94684,     19907,       970,        1;
  7:  2908,  176253,  1229514,   1904735,    814312,    104115,     2914,       1;
  8:  8739,  954906, 11423702,  31353078,  25176748,   6488710,   534250,    8746,     1;
  9: 26234, 5030841, 98126944, 444857368, 619473732, 294494902, 49391008, 2712168, 26242, 1;

%F Formula

The row polynomials satisfy 16*P(n,x) = (48*n^2-128*n+52+(24*n^2-64*n-70)*x-112*x^2-(48*n^2-176*n+108)*x^3)*P(n-4,x) + ((24*n^2-64*n+26)*x-48*x^2-(48*n^2-176*n+108)*x^3)*d/dx P(n-4,x) + (48*n^3-284*n^2+468*n-169+(24*n^3-76*n^2-134*n+475)*x-(24*n^2+88*n-418)*x^2-(48*n^3-296*n^2+548*n-142)*x^3)*P(n-3,x) + (60*n^2-160*n+65+(24*n^3-196*n^2+378*n-263)*x-(48*n^2-104*n-184)*x^2-(48*n^3-344*n^2+724*n-426)*x^3+(48*n^2-176*n+108)*x^4)*d/dx P(n-3,x) + ((24*n^2-64*n+26)*x-(24*n^2-64*n+74)*x^2-(48*n^2-176*n+60)*x^3+(48*n^2-176*n+108)*x^4)*d^2/dx^2 P(n-3,x) - (12*n^2+40*n-23+(96*n^2-464*n+200)*x+(40*n^2-200*n+298)*x^2+(128*n-192)*x^3)*P(n-2,x) - (12*n^2-32*n+13+(48*n-208)*x-(24*n^2-40*n-226)*x^2+32*x^3-128*x^4)*d/dx P(n-2,x) + (64*x^2-128*x^3+64*x^4)*d^2/dx^2 P(n-2,x) + (24*n+36+(64*n-208)*x+64*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 9 + 10*x + x^2, P(3,x) = 32 + 93*x + 34*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 9 + 10*x + x^2;
P[3, x_] := 32 + 93*x + 34*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((48*n^2 - 128*n + 52 + (24*n^2 - 64*n - 70)*x - 112*x^2 - (48*n^2 - 176*n + 108)*x^3)*P[n - 4, x] + ((24*n^2 - 64*n + 26)*x - 48*x^2 - (48*n^2 - 176*n + 108)*x^3)*D[P[n - 4, x], x] + (48*n^3 - 284*n^2 + 468*n - 169 + (24*n^3 - 76*n^2 - 134*n + 475)*x - (24*n^2 + 88*n - 418)*x^2 - (48*n^3 - 296*n^2 + 548*n - 142)*x^3)*P[n - 3, x] + (60*n^2 - 160*n + 65 + (24*n^3 - 196*n^2 + 378*n - 263)*x - (48*n^2 - 104*n - 184)*x^2 - (48*n^3 - 344*n^2 + 724*n - 426)*x^3 + (48*n^2 - 176*n + 108)*x^4)*D[P[n - 3, x], x] + ((24*n^2 - 64*n + 26)*x - (24*n^2 - 64*n + 74)*x^2 - (48*n^2 - 176*n + 60)*x^3 + (48*n^2 - 176*n + 108)*x^4)*D[P[n - 3, x], {x, 2}] - (12*n^2 + 40*n - 23 + (96*n^2 - 464*n + 200)*x + (40*n^2 - 200*n + 298)*x^2 + (128*n - 192)*x^3)*P[n - 2, x] - (12*n^2 - 32*n + 13 + (48*n - 208)*x - (24*n^2 - 40*n - 226)*x^2 + 32*x^3 - 128*x^4)*D[P[n - 2, x], x] + (64*x^2 - 128*x^3 + 64*x^4)*D[P[n - 2, x], {x, 2}] + (24*n + 36 + (64*n - 208)*x + 64*x^2)*P[n - 1, x])/(16)]];
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] = 2 + x; v[3] = 9 + 10*x + x^2; v[4] = 32 + 93*x + 34*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((48*n^2 - 128*n + 52 + (24*n^2 - 64*n - 70)*x - 112*x^2 - (48*n^2 - 176*n + 108)*x^3)*v[i - 4] + ((24*n^2 - 64*n + 26)*x - 48*x^2 - (48*n^2 - 176*n + 108)*x^3)*deriv(v[i - 4], x) + (48*n^3 - 284*n^2 + 468*n - 169 + (24*n^3 - 76*n^2 - 134*n + 475)*x - (24*n^2 + 88*n - 418)*x^2 - (48*n^3 - 296*n^2 + 548*n - 142)*x^3)*v[i - 3] + (60*n^2 - 160*n + 65 + (24*n^3 - 196*n^2 + 378*n - 263)*x - (48*n^2 - 104*n - 184)*x^2 - (48*n^3 - 344*n^2 + 724*n - 426)*x^3 + (48*n^2 - 176*n + 108)*x^4)*deriv(v[i - 3], x) + ((24*n^2 - 64*n + 26)*x - (24*n^2 - 64*n + 74)*x^2 - (48*n^2 - 176*n + 60)*x^3 + (48*n^2 - 176*n + 108)*x^4)*deriv(deriv(v[i - 3], x), x) - (12*n^2 + 40*n - 23 + (96*n^2 - 464*n + 200)*x + (40*n^2 - 200*n + 298)*x^2 + (128*n - 192)*x^3)*v[i - 2] - (12*n^2 - 32*n + 13 + (48*n - 208)*x - (24*n^2 - 40*n - 226)*x^2 + 32*x^3 - 128*x^4)*deriv(v[i - 2], x) + (64*x^2 - 128*x^3 + 64*x^4)*deriv(deriv(v[i - 2], x), x) + (24*n + 36 + (64*n - 208)*x + 64*x^2)*v[i - 1])/(16)); 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 T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.