A013612 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(11 + 60\,x + 25\,x^{2} + 5\,n + 25\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 11 + 60\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	11	60	25															96
1	16	165	450	125														756
2	21	320	1650	3000	625													5616
3	26	525	4000	13750	18750	3125												40176
4	31	780	7875	40000	103125	112500	15625											279936
5	36	1085	13650	91875	350000	721875	656250	78125										1912896
6	41	1440	21700	182000	918750	2800000	4812500	3750000	390625									12877056
7	46	1845	32400	325500	2047500	8268750	21000000	30937500	21093750	1953125								85660416
8	51	2300	46125	540000	4068750	20475000	68906250	150000000	193359375	117187500	9765625							564350976
9	56	2805	63250	845625	7425000	44756250	187687500	541406250	1031250000	1181640625	644531250	48828125						3688436736
10	61	3360	84150	1265000	12684375	89100000	447562500	1608750000	4060546875	6875000000	7089843750	3515625000	244140625					23944605696
11	66	3965	109200	1823250	20556250	164896875	965250000	4155937500	13071093750	29326171875	44687500000	41894531250	19042968750	1220703125				154551545856
12	71	4620	138775	2548000	31906875	287787500	1923796875	9652500000	36364453125	101664062500	205283203125	284375000000	244384765625	102539062500	6103515625			992612745216
13	76	5325	173250	3469375	47775000	478603125	3597343750	20612109375	90492187500	303037109375	762480468750	1399658203125	1777343750000	1409912109375	549316406250	30517578125		6347497291776
14	81	6080	213000	4620000	69387500	764400000	6381375000	41112500000	206121093750	804375000000	2424296875000	5545312500000	9331054687500	10937500000000	8056640625000	2929687500000	152587890625	40435908673536

Technical details

Node id: n-cbf6b57b9edf0124 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f93b0ee1ce1d2e74

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 11,   60,    25;
  1: 16,  165,   450,    125;
  2: 21,  320,  1650,   3000,     625;
  3: 26,  525,  4000,  13750,   18750,     3125;
  4: 31,  780,  7875,  40000,  103125,   112500,     15625;
  5: 36, 1085, 13650,  91875,  350000,   721875,    656250,     78125;
  6: 41, 1440, 21700, 182000,  918750,  2800000,   4812500,   3750000,     390625;
  7: 46, 1845, 32400, 325500, 2047500,  8268750,  21000000,  30937500,   21093750,    1953125;
  8: 51, 2300, 46125, 540000, 4068750, 20475000,  68906250, 150000000,  193359375,  117187500,   9765625;
  9: 56, 2805, 63250, 845625, 7425000, 44756250, 187687500, 541406250, 1031250000, 1181640625, 644531250, 48828125;

%F Formula

The row polynomials satisfy (5*n+6+5*x)*P(n,x) = (5*n+11+(25*n+60)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 11 + 60*x + 25*x^2.

%t Mathematica

P[0, x_] := 11 + 60*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 11 + (25*n + 60)*x + 25*x^2)*P[n - 1, x])/(5*n + 6 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 11 + 60*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5*n + 11 + (25*n + 60)*x + 25*x^2)*v[i - 1])/(5*n + 6 + 5*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 of coefficients in expansion of (1+5x)^n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.