A128813 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 16\,n x - n^{2} - 8\,n^{2} x + 36\,n^{2} x^{2} - 3\,n^{3} - 8\,n^{3} x - 36\,n^{3} x^{2} + n^{4} + 8\,n^{4} x - 36\,n^{4} x^{2} - n^{5} - 8\,n^{5} x + 36\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-16\,n x^{2} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x^{2} - 16\,n^{3} x^{3} - 8\,n^{4} x^{2} - 16\,n^{4} x^{3} + 8\,n^{5} x^{2} + 16\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(4\,n - 16\,n x + 4\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(16\,n x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	8	80	108													196
3	56	944	2916	2304												6220
4	616	14864	74556	129024	72000											291060
5	9856	311744	2196216	6040704	7200000	3110400										18868920
6	216832	8514176	77776560	296879616	554821200	503884800	177811200									1619904384
7	6288128	295481472	3328306416	16352597504	42066781200	59353440768	43563744000	13005619200								177972258688
8	232660736	12743795328	171015335856	1031069328896	3396138123600	6557561183232	7428495723264	4577977958400	1185137049600							24360371158912
9	10702393856	669972450048	10447324003296	74791642866176	301216478061600	741787315247232	1145012048216064	1083814238453760	575976606105600	131681894400000						4065408222197632
10	599334055936	42227510499328	750868325571456	6231364250039296	29722936389073600	89252779778802432	175182719955171264	225201450811392000	183141953451648000	85593231360000000	17526860144640000					812646991310893312
11	40155381747712	3145691584989184	62849748431587968	593705171820067840	3276655614638536000	11629671994014872832	27772991670587978304	45291428187654832128	49893228257438592000	35580101878425600000	14845250542510080000	2753310393630720000				191702379306119603968
12	3172275158069248	273566593424717824	6069267872426653056	64333038805479089152	403571429187585880000	1654812004799575680768	4663433362038941617344	9236911578644127154176	12883791385137749686272	12419894231309721600000	7888874823429912576000	2973575225121177600000	504085244567224320000			52699628329782504643840
13	291849314542370816	27477542910148452352	670398164270674030080	7882384908382566612992	55423279395566016937600	258010704603043470061056	838969576496544182159232	1958360595868516676935680	3312973658875981135306752	4037506852113914687078400	3460885591374578847744000	1982260089078513205248000	681523250654887280640000	106400762391727964160000		16700894913318151397736960
14	30936027341491306496	3157772972691327434752	84045344437736590924800	1085814781616270365540352	8461275163993027465273600	44109134377141771348402176	162678834841003608501330432	437703592715400600576000000	871670734961533198384326912	1286894823069699695613542400	1392784363807851079847485440	1075054943683374171291648000	560785982356893285457920000	177263670144618788290560000	25650183790862991360000000	6044230587747426257911695360

Technical details

Node id: n-a148d5351dbfaee8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9b3fddac6ffcbf8c

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,            8;
  2:           8,           80,            108;
  3:          56,          944,           2916,           2304;
  4:         616,        14864,          74556,         129024,           72000;
  5:        9856,       311744,        2196216,        6040704,         7200000,         3110400;
  6:      216832,      8514176,       77776560,      296879616,       554821200,       503884800,        177811200;
  7:     6288128,    295481472,     3328306416,    16352597504,     42066781200,     59353440768,      43563744000,      13005619200;
  8:   232660736,  12743795328,   171015335856,  1031069328896,   3396138123600,   6557561183232,    7428495723264,    4577977958400,   1185137049600;
  9: 10702393856, 669972450048, 10447324003296, 74791642866176, 301216478061600, 741787315247232, 1145012048216064, 1083814238453760, 575976606105600, 131681894400000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(8*n^5-8*n^4+8*n^3+8*n^2-16*n)*x-(36*n^5-36*n^4-36*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^5-8*n^4+8*n^3+8*n^2-16*n)*x^2+(16*n^5-16*n^4-16*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(16*n^3-16*n)*x)*P(n-1,x) - (16*n^3-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x - (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x^2 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (16*n^3 - 16*n)*x)*P[n - 1, x] - (16*n^3 - 16*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x - (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^5 - 8*n^4 + 8*n^3 + 8*n^2 - 16*n)*x^2 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (16*n^3 - 16*n)*x)*v[i - 1] - (16*n^3 - 16*n)*x^2*deriv(v[i - 1], x))/(4*n - 4)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
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 basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.