A272866 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - x^{2} + 5\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(25 + 50\,x + 35\,x^{2} + 10\,x^{3} + x^{4} + 25\,n x + 35\,n x^{2} + 15\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	5	10	3																											18
2	25	100	105	40	5																									275
3	125	750	1350	1100	450	90	7																							3872
4	625	5000	12750	16000	11375	4800	1190	160	9																					51909
5	3125	31250	103125	175000	178125	116250	49875	14000	2475	250	11																			673486
6	15625	187500	759375	1625000	2156250	1912500	1176875	510000	155250	32500	4455	360	13																	8535703
7	78125	1093750	5250000	13562500	22203125	24937500	20059375	11825000	5158125	1662500	390775	65100	7280	490	15															106293660
8	390625	6250000	34687500	105000000	204531250	278250000	276850000	207000000	118096875	51750000	17402000	4452000	850850	117600	11100	640	17													1305640457
9	1953125	35156250	221484375	768750000	1736718750	2775937500	3293718750	2988000000	2110809375	1173593750	515975625	179280000	48935250	10363500	1667250	196800	16065	810	19											15862557194
10	9765625	195312500	1376953125	5390625000	13857421875	25453125000	35092968750	37462500000	31596328125	21329687500	11614521875	5119125000	1825565625	524475000	120318750	21720000	3015375	310500	22325	1000	21									190993762971
11	48828125	1074218750	8378906250	36523437500	105273437500	218496093750	343341796875	421781250000	413827734375	329011718750	214003625000	114543187500	50582675000	18424656250	5517703125	1349700000	266825625	41951250	5120500	467500	30030	1210	23							2282493364888
12	244140625	5859375000	50097656250	240625000000	768603515625	1778906250000	3138652343750	4362187500000	4883097656250	4471328125000	3385372031250	2134822500000	1126269015625	498125250000	184656656250	57233000000	14757806250	3140775000	545228750	75900000	8264025	677600	39330	1440	25					27104608708045
13	1220703125	31738281250	295166015625	1548828125000	5430419921875	13867089843750	27146259765625	42117968750000	52988203125000	54937695312500	47487395312500	34505371875000	21195861171875	11045777968750	4891352578125	1840286500000	587116887500	158220562500	35796475000	6738875000	1042416375	130165750	12789725	951600	50375	1690	27			320119694425542
14	6103515625	170898437500	1717529296875	9775390625000	37324218750000	104241210937500	224225244140625	384200781250000	536756923828125	621714843750000	604357660156250	497474250000000	349004870703125	209616717187500	108087336328125	47912392500000	18255639390625	5969691000000	1670224806250	397897500000	80155700625	13523867500	1886054625	213486000	19110000	1301300	63315	1960	29	3762975623688479

Technical details

Node id: n-5bd8ef4eded0ab6c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6a35350bf0e37fe8

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,         3;
  2:      25,      100,       105,        40,          5;
  3:     125,      750,      1350,      1100,        450,         90,          7;
  4:     625,     5000,     12750,     16000,      11375,       4800,       1190,        160,          9;
  5:    3125,    31250,    103125,    175000,     178125,     116250,      49875,      14000,       2475,        250,        11;
  6:   15625,   187500,    759375,   1625000,    2156250,    1912500,    1176875,     510000,     155250,      32500,      4455,       360,       13;
  7:   78125,  1093750,   5250000,  13562500,   22203125,   24937500,   20059375,   11825000,    5158125,    1662500,    390775,     65100,     7280,      490,      15;
  8:  390625,  6250000,  34687500, 105000000,  204531250,  278250000,  276850000,  207000000,  118096875,   51750000,  17402000,   4452000,   850850,   117600,   11100,    640,    17;
  9: 1953125, 35156250, 221484375, 768750000, 1736718750, 2775937500, 3293718750, 2988000000, 2110809375, 1173593750, 515975625, 179280000, 48935250, 10363500, 1667250, 196800, 16065, 810, 19;

%F Formula

The row polynomials satisfy (5+5*n*x+(2*n-1)*x^2)*P(n,x) = (25+(25*n+50)*x+(35*n+35)*x^2+(15*n+10)*x^3+(2*n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(5 + 5*n*x + (2*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((25 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(5 + 5*n*x + (2*n - 1)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.