A257615 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 2\,n\right)\,P_{n}(x) = \left(-75 - 30\,x - 15\,x^{2} - 30\,n - 72\,n x - 46\,n x^{2} - 24\,n^{2} x - 36\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 30\,x^{3} - 12\,n x - 20\,n x^{2} + 32\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(30 + 18\,x + 16\,n + 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 10\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	10														15
2	25	140	75													240
3	125	1430	2145	500												4200
4	625	12760	38610	25520	3125											80640
5	3125	105570	558030	744040	263925	18750										1693440
6	15625	832740	7080885	16368880	11801475	2498220	109375									38707200
7	78125	6360430	82464615	302793020	378491275	164929230	22261505	625000								958003200
8	390625	47491760	904372500	4980015520	9840773150	7470023280	2110202500	189967040	3515625							25546752000
9	1953125	348848570	9492867660	75279280720	221305341950	265566410340	131738741260	25314313760	1569818565	19531250						730637107200
10	9765625	2531783740	96424538895	1068558068640	4476654160650	7966992310200	6267315824910	2137116137280	289273616685	12658918700	107421875					22317642547200
11	48828125	18210767430	955167389085	14454025844100	83574758218650	210828629530260	245966734451970	133719613149840	32521558149225	3183891296950	100159220865	585937500				725323382784000
12	244140625	130112090760	9279410280390	188286084217040	1465890035250075	5067933930339120	8362868971366980	6757245240452160	2638602063450135	470715210542600	34024504361430	780672544560	3173828125			24995759652864000
13	1220703125	924944791570	88784232199290	2380460602400440	24469795379490875	112959437843388690	254507974635755820	290866256726578080	169439156765083035	48939590758981750	6546266656601210	355136928797160	6012141145205	17089843750		910559815925760000
14	6103515625	6550297134740	839293188226905	29381298985579280	392496733758395125	2369757920124778500	7094141795970871845	11052917755609967040	9121039451962549515	3949596533541297500	863492814268469275	88143896956737840	3636937148983255	45852079943180	91552734375	34965496931549184000

Technical details

Node id: n-baf012a19aece82b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-27bcff2c51d51bee

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:       5,        10;
  2:      25,       140,         75;
  3:     125,      1430,       2145,         500;
  4:     625,     12760,      38610,       25520,         3125;
  5:    3125,    105570,     558030,      744040,       263925,        18750;
  6:   15625,    832740,    7080885,    16368880,     11801475,      2498220,       109375;
  7:   78125,   6360430,   82464615,   302793020,    378491275,    164929230,     22261505,      625000;
  8:  390625,  47491760,  904372500,  4980015520,   9840773150,   7470023280,   2110202500,   189967040,    3515625;
  9: 1953125, 348848570, 9492867660, 75279280720, 221305341950, 265566410340, 131738741260, 25314313760, 1569818565, 19531250;

%F Formula

The row polynomials satisfy (2*n+3)*P(n,x) = -(30*n+75+(24*n^2+72*n+30)*x+(8*n^3+36*n^2+46*n+15)*x^2)*P(n-2,x) - ((12*n+30)*x+(8*n^2+20*n)*x^2-(8*n^2+32*n+30)*x^3)*d/dx P(n-2,x) + (16*n+30+(8*n^2+24*n+18)*x)*P(n-1,x) + ((4*n+6)*x-(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 75 + (24*n^2 + 72*n + 30)*x + (8*n^3 + 36*n^2 + 46*n + 15)*x^2)*P[n - 2, x] - ((12*n + 30)*x + (8*n^2 + 20*n)*x^2 - (8*n^2 + 32*n + 30)*x^3)*D[P[n - 2, x], x] + (16*n + 30 + (8*n^2 + 24*n + 18)*x)*P[n - 1, x] + ((4*n + 6)*x - (4*n + 6)*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];
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] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 75 + (24*n^2 + 72*n + 30)*x + (8*n^3 + 36*n^2 + 46*n + 15)*x^2)*v[i - 2] - ((12*n + 30)*x + (8*n^2 + 20*n)*x^2 - (8*n^2 + 32*n + 30)*x^3)*deriv(v[i - 2], x) + (16*n + 30 + (8*n^2 + 24*n + 18)*x)*v[i - 1] + ((4*n + 6)*x - (4*n + 6)*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
    
    ↑ 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.