A142462 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 84\,n + 49\,n^{2}\right)\,P_{n}(x) = \left(-36 - 15600\,x + 2340\,x^{2} - 210\,n + 6580\,n x - 36204\,n x^{2} + 294\,n^{2} + 3528\,n^{2} x + 90797\,n^{2} x^{2} - 1372\,n^{3} x - 78890\,n^{3} x^{2} + 21609\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2436\,x - 13380\,x^{2} + 3420\,x^{3} + 490\,n x - 3787\,n x^{2} - 44772\,n x^{3} - 686\,n^{2} x + 7448\,n^{2} x^{2} + 97559\,n^{2} x^{3} + 1372\,n^{3} x^{2} - 66542\,n^{3} x^{3} + 9604\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2352\,x^{2} - 3108\,x^{3} + 756\,x^{4} + 1715\,n x^{2} + 7105\,n x^{3} - 8820\,n x^{4} - 2401\,n^{2} x^{2} - 15092\,n^{2} x^{3} + 17493\,n^{2} x^{4} + 9604\,n^{3} x^{3} - 9604\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(72 - 864\,x + 126\,n + 3024\,n x - 245\,n^{2} - 3528\,n^{2} x + 1372\,n^{3} x\right)\,P_{n-1}(x) + \left(-84\,x - 144\,x^{2} - 833\,n x - 672\,n x^{2} + 686\,n^{2} x + 2156\,n^{2} x^{2} - 1372\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	64	9													74
3	1	572	1287	16												1876
4	1	4664	38610	18656	25											61956
5	1	37428	810018	1440032	233925	36										2521440
6	1	299568	14592447	63361408	40534575	2696112	49									121484160
7	1	2396716	243151713	2146272688	3353551075	972606852	29359771	64								6747338880
8	1	19173928	3879157968	62779708768	194505962350	141254344728	21119860048	306782848	81							423864990720
9	1	153391652	60344436420	1677693668672	9172031526350	13207725397944	5137936860308	429115992320	3106180953	100						29688107454720
10	1	1227133472	924839025669	42273210614784	378708957628050	950956228651968	742269556950978	169092842459136	8323551231021	30678336800	121					2292581092032000
11	1	9817068060	14049292605003	1023727654793040	14285154350493450	57684082887799704	78514446152838486	36569995137263232	5182621252389765	156103251166700	296966308815	144				193430486762726400
12	1	78536544792	212307665697630	24120505697393888	505445220419315175	3105158097196317744	6752242369144109796	5520281061682342656	1637642514158581167	150753160608711800	2854358616601470	2827315612512	169			17698712501041228800
13	1	628292358676	3198398149075446	557449070701833056	17069961961899528875	153272635191084828348	501669681265723901028	655242032673598564608	344863429179940863783	68279847847598115500	4215708597182612486	51174370385207136	26545352154061	196		1745225145729909043200
14	1	5026338869776	48096133149728475	12707390765445455744	556870903176070942025	7090982561287714401648	33463198241334459010203	65524203267359856460800	55316715460165126118907	19697173781354762226800	2695255171372183359401	114366516889009101696	903138500256012475	246290604619024	225	184472424779654976307200

Technical details

Node id: n-5ed35238ff3803e6 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9b672623d94ceedb

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: 1,         4;
  2: 1,        64,           9;
  3: 1,       572,        1287,            16;
  4: 1,      4664,       38610,         18656,            25;
  5: 1,     37428,      810018,       1440032,        233925,             36;
  6: 1,    299568,    14592447,      63361408,      40534575,        2696112,            49;
  7: 1,   2396716,   243151713,    2146272688,    3353551075,      972606852,      29359771,           64;
  8: 1,  19173928,  3879157968,   62779708768,  194505962350,   141254344728,   21119860048,    306782848,         81;
  9: 1, 153391652, 60344436420, 1677693668672, 9172031526350, 13207725397944, 5137936860308, 429115992320, 3106180953, 100;

%F Formula

The row polynomials satisfy (49*n^2-84*n+36)*P(n,x) = (294*n^2-210*n-36-(1372*n^3-3528*n^2-6580*n+15600)*x+(21609*n^4-78890*n^3+90797*n^2-36204*n+2340)*x^2)*P(n-2,x) - ((686*n^2-490*n-2436)*x-(1372*n^3+7448*n^2-3787*n-13380)*x^2-(9604*n^4-66542*n^3+97559*n^2-44772*n+3420)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1715*n-2352)*x^2-(9604*n^3-15092*n^2+7105*n-3108)*x^3+(9604*n^3-17493*n^2+8820*n-756)*x^4)*d^2/dx^2 P(n-2,x) - (245*n^2-126*n-72-(1372*n^3-3528*n^2+3024*n-864)*x)*P(n-1,x) + ((686*n^2-833*n-84)*x-(1372*n^3-2156*n^2+672*n+144)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*P[n - 2, x] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*D[P[n - 2, x], {x, 2}] - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*P[n - 1, x] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 84*n + 36)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*v[i - 2] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*deriv(deriv(v[i - 2], x), x) - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*v[i - 1] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 84*n + 36)); 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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-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.