A013612 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 150\,x + 400\,x^{2} + 90\,n x + 400\,n x^{2} + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 1620\,x + 10200\,x^{2} + 18000\,x^{3} + 540\,n x + 5400\,n x^{2} + 12000\,n x^{3} + 600\,n^{2} x^{2} + 2000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 36 + 960\,x + 3600\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	36	960	3600															4596
1	216	8640	64800	128000														201656
2	1296	69120	777600	3072000	4000000													7920016
3	7776	518400	7776000	46080000	120000000	115200000												289582176
4	46656	3732480	69984000	552960000	2160000000	4147200000	3136000000											10069923136
5	279936	26127360	587865600	5806080000	30240000000	87091200000	131712000000	81920000000										337383552896
6	1679616	179159040	4702924800	55738368000	362880000000	1393459200000	3161088000000	3932160000000	2073600000000									10983809331456
7	10077696	1209323520	36279705600	501645312000	3919104000000	18811699200000	56899584000000	106168320000000	111974400000000	51200000000000								349512251618816
8	60466176	8062156800	272097792000	4299816960000	39191040000000	225740390400000	853493760000000	2123366400000000	3359232000000000	3072000000000000	1239040000000000							10916643627774976
9	362797056	53210234880	1995383808000	35473489920000	369515520000000	2483144294400000	11266117632000000	35035545600000000	73903104000000000	101376000000000000	81776640000000000	29491200000000000						335738789493159936
10	2176782336	348285173760	14366763417600	283787919360000	3325639680000000	25540912742400000	135193411584000000	504511856640000000	1330255872000000000	2433024000000000000	2943959040000000000	2123366400000000000	692224000000000000					10191699637791133696
11	13060694016	2263853629440	101873413324800	2213545771008000	28822210560000000	249023899238400000	1506440871936000000	6558654136320000000	20751991603200000000	47443968000000000000	76542935040000000000	82811289600000000000	53993472000000000000	16056320000000000000				305945235057353056256
12	78364164096	14627977297920	713113893273600	16903440433152000	242106568704000000	2324223059558400000	15817629155328000000	78703849635840000000	290527882444800000000	797058662400000000000	1607401635840000000000	2318716108800000000000	2267725824000000000000	1348730880000000000000	368640000000000000000			9095906433164898287616
13	470184984576	94036996915200	4936942338048000	126775803248640000	1980871925760000000	20918007536025600000	158176291553280000000	885418308403200000000	3735358488576000000000	11955879936000000000000	28933229445120000000000	52171112448000000000000	68031774720000000000000	60692889600000000000000	33177600000000000000000	8388608000000000000000		268153077924367034187776
14	2821109907456	601836780257280	33853318889472000	936190547066880000	15846975406080000000	182557156678041600000	1518492398911488000000	9444461956300800000000	44824301862912000000000	163966353408000000000000	462931671121920000000000	1001685359001600000000000	1632762593280000000000000	1942172467200000000000000	1592524800000000000000000	805306368000000000000000	189399040000000000000000	7846735283010252256116736

Technical details

Node id: n-a6b0708a7afa5887 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-40433d4e786667ae

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        36,         960,          3600;
  1:       216,        8640,         64800,         128000;
  2:      1296,       69120,        777600,        3072000,         4000000;
  3:      7776,      518400,       7776000,       46080000,       120000000,        115200000;
  4:     46656,     3732480,      69984000,      552960000,      2160000000,       4147200000,        3136000000;
  5:    279936,    26127360,     587865600,     5806080000,     30240000000,      87091200000,      131712000000,       81920000000;
  6:   1679616,   179159040,    4702924800,    55738368000,    362880000000,    1393459200000,     3161088000000,     3932160000000,     2073600000000;
  7:  10077696,  1209323520,   36279705600,   501645312000,   3919104000000,   18811699200000,    56899584000000,   106168320000000,   111974400000000,     51200000000000;
  8:  60466176,  8062156800,  272097792000,  4299816960000,  39191040000000,  225740390400000,   853493760000000,  2123366400000000,  3359232000000000,   3072000000000000,  1239040000000000;
  9: 362797056, 53210234880, 1995383808000, 35473489920000, 369515520000000, 2483144294400000, 11266117632000000, 35035545600000000, 73903104000000000, 101376000000000000, 81776640000000000, 29491200000000000;

%F Formula

The row polynomials satisfy (9+(90*n+150)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (54+(540*n+1620)*x+(600*n^2+5400*n+10200)*x^2+(2000*n^2+12000*n+18000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 960*x + 3600*x^2.

%t Mathematica

P[0, x_] := 36 + 960*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*P[n - 1, x])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 960*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*v[i - 1])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*x^2)); 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_A013611 of Triangle of coefficients in expansion of (1+5x)^n.?
  - Geometric basis map (elementary) 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.