A257621 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,n\right)\,P_{n}(x) = \left(9 - 30\,x - 15\,x^{2} + 12\,n - 16\,n x + 52\,n x^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 12\,x^{3} + 16\,n x - 48\,n x^{2} + 32\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 2\,x + 8\,n - 16\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	84	27													120
3	27	786	1179	108												2100
4	81	6312	25938	12624	405											45360
5	243	47262	427338	569784	118155	1458										1164240
6	729	342012	6047685	17093520	10079475	1026036	5103									34594560
7	2187	2433450	78323949	409610820	512013525	156647898	8517075	17496								1167566400
8	6561	17169744	960118164	8546096736	19456513950	12819145104	2240275716	68678976	59049							44108064000
9	19683	120647478	11359692900	162755704944	615374046210	738448855452	284822483652	30292514400	542913651	196830						1843717075200
10	59049	846067620	131290614495	2910869547360	17185111919850	33968647350792	24059156687790	5821739094720	393871843485	4230338100	649539					84475764172800
11	177147	5927551554	1493691715215	49789938553500	439313321437350	1338076515271356	1561089267816582	702901314299760	112027361745375	4978972384050	32601533547	2125764				4209708914611200
12	531441	41509512696	16812935942598	824521047493680	10525262919025275	47118215408710608	84298820462095428	62824287211614144	18945473254245495	2061302618734200	61647431789526	249057076176	6908733			226676633863680000
13	1594323	290620795854	187868716013646	13331757373114152	240175396526797125	1525779996999405534	3980176943091610212	4548773649247554528	2288669995499108301	480350793053594250	36662332776063918	751474864054584	1889035173051	22320522		13114862387827200000
14	4782969	2034520946508	2088788367032937	211749466800234096	5279914492814031045	46333148488440432732	169767460674282535029	282023966253348380736	218272449438363259323	77221914147400721220	11615811884190868299	635248400400702288	9051416257142727	14241646625556	71744535	811372819726909440000

Technical details

Node id: n-40ab481764296b44 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-ca6524666417859c

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:     3,         6;
  2:     9,        84,          27;
  3:    27,       786,        1179,          108;
  4:    81,      6312,       25938,        12624,          405;
  5:   243,     47262,      427338,       569784,       118155,         1458;
  6:   729,    342012,     6047685,     17093520,     10079475,      1026036,         5103;
  7:  2187,   2433450,    78323949,    409610820,    512013525,    156647898,      8517075,       17496;
  8:  6561,  17169744,   960118164,   8546096736,  19456513950,  12819145104,   2240275716,    68678976,     59049;
  9: 19683, 120647478, 11359692900, 162755704944, 615374046210, 738448855452, 284822483652, 30292514400, 542913651, 196830;

%F Formula

The row polynomials satisfy (4*n-1)*P(n,x) = (12*n+9+(32*n^2-16*n-30)*x-(64*n^3-48*n^2-52*n+15)*x^2)*P(n-2,x) + ((16*n+12)*x-(64*n^2+48*n)*x^2+(64*n^2+32*n-12)*x^3)*d/dx P(n-2,x) + (8*n-6+(32*n^2-16*n+2)*x)*P(n-1,x) + ((16*n-4)*x-(16*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*P[n - 2, x] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*D[P[n - 2, x], x] + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*P[n - 1, x] + ((16*n - 4)*x - (16*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n - 1)]];
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] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*v[i - 2] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*deriv(v[i - 2], x) + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*v[i - 1] + ((16*n - 4)*x - (16*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n - 1)); 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 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

[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.