A272866 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 - 12\,x + 16\,x^{2} + 20\,n + 32\,n x\right)\,P_{n}(x) = \left(25 + 200\,x + 560\,x^{2} + 640\,x^{3} + 256\,x^{4} + 100\,n + 560\,n x + 960\,n x^{2} + 512\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	25	52	16																											93
2	225	1320	2480	1664	256																									5945
3	1625	15900	60000	109760	99840	39936	4096																							331157
4	10625	146000	836000	2585600	4678400	4997120	2990080	851968	65536																					17161329
5	65625	1162500	8950000	39280000	108320000	194944000	229888000	172851200	77004800	17039360	1048576																			850554061
6	390625	8475000	82050000	467600000	1742400000	4458240000	8000000000	10088448000	8798208000	5098700800	1808793600	327155712	16777216																	40897238953
7	2265625	58187500	679000000	4764200000	22416800000	74681600000	181260800000	324700160000	429793280000	416088064000	287545753600	135496990720	39929774080	6106906624	268435456															1923792217605
8	12890625	382500000	5225000000	43568000000	247912000000	1019468800000	3129548800000	7303577600000	13069312000000	17945067520000	18775801856000	14732912230400	8426356736000	3344705781760	842887331840	111669149696	4294967296													89002703163617
9	72265625	2432812500	38081250000	367980000000	2457720000000	12037536000000	44763264000000	129082982400000	292255948800000	522619289600000	738477342720000	820506525696000	708959010816000	467389749657600	227941967462400	78146929950720	17201344020480	2010044694528	68719476736											4064326941622589
10	400390625	15078125000	266093750000	2924250000000	22429500000000	127550400000000	557596800000000	1917342720000000	5262382080000000	11630698496000000	20789057945600000	30061179699200000	35038848614400000	32658696437760000	24017255792640000	13646721712128000	5799816462336000	1749769676390400	341879396761600	35734127902720	1099511627776									183661265195012121
11	2197265625	91523437500	1797812500000	22144375000000	191840000000000	1242714000000000	6247560000000000	24971020800000000	80637004800000000	212623994880000000	460793118720000000	823392482099200000	1213419067801600000	1470722421555200000	1457395649740800000	1169171970785280000	748089350553600000	373398014263296000	140600049401856000	37871303629209600	6652045348044800	628920651087872	17592186044416							8228090157123421813
12	11962890625	546093750000	11817187500000	161260000000000	1557352500000000	11322960000000000	64372880000000000	293394816000000000	1090476288000000000	3344322150400000000	8530630901760000000	18188320899072000000	32494373306368000000	48651592689254400000	60917183781273600000	63496648864563200000	54688919165337600000	38493515541381120000	21792062764482560000	9696867923263488000	3277341796702617600	797277871532933120	127191505101127680	10977524091715584	281474976710656					365968806091305213265
13	64697265625	3211914062500	75917968750000	1136931250000000	12111287500000000	97673290000000000	619687640000000000	3173044160000000000	13347276800000000000	46707561472000000000	137192257945600000000	340313616547840000000	715676208332800000000	1278426125172736000000	1940027210727424000000	2496815141853593600000	2715529204308377600000	2481457405834035200000	1889218674556928000000	1184188797802250240000	601031950966194176000	241238238741030502400	73955351107469312000	16402250580371701760	2396759426691235840	190277084256403456	4503599627370496			16178022545362555420077
14	347900390625	18662109375000	478378906250000	7800406250000000	90863500000000000	805013300000000000	5639106200000000000	32055200320000000000	150589479040000000000	592402370560000000000	1970288832512000000000	5578799946137600000000	13512754076057600000000	28084970884956160000000	50164491710627840000000	77010521903071232000000	101473013903392768000000	114433776335768780800000	109945463785494937600000	89404018617692979200000	60971793477082808320000	34440795464711274496000	15839395184950378496000	5789774904389376409600	1623941730634458726400	330935759618252472320	44608154309104762880	3278620528725721088	72057594037927936	711404285680245480947849

Technical details

Node id: n-9cbd345833a8bbca · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-72d6e0ea460d9703

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:       25,         52,          16;
  2:      225,       1320,        2480,         1664,           256;
  3:     1625,      15900,       60000,       109760,         99840,          39936,           4096;
  4:    10625,     146000,      836000,      2585600,       4678400,        4997120,        2990080,          851968,           65536;
  5:    65625,    1162500,     8950000,     39280000,     108320000,      194944000,      229888000,       172851200,        77004800,        17039360,         1048576;
  6:   390625,    8475000,    82050000,    467600000,    1742400000,     4458240000,     8000000000,     10088448000,      8798208000,      5098700800,      1808793600,       327155712,        16777216;
  7:  2265625,   58187500,   679000000,   4764200000,   22416800000,    74681600000,   181260800000,    324700160000,    429793280000,    416088064000,    287545753600,    135496990720,     39929774080,      6106906624,       268435456;
  8: 12890625,  382500000,  5225000000,  43568000000,  247912000000,  1019468800000,  3129548800000,   7303577600000,  13069312000000,  17945067520000,  18775801856000,  14732912230400,   8426356736000,   3344705781760,    842887331840,   111669149696,     4294967296;
  9: 72265625, 2432812500, 38081250000, 367980000000, 2457720000000, 12037536000000, 44763264000000, 129082982400000, 292255948800000, 522619289600000, 738477342720000, 820506525696000, 708959010816000, 467389749657600, 227941967462400, 78146929950720, 17201344020480, 2010044694528, 68719476736;

%F Formula

The row polynomials satisfy (20*n-15+(32*n-12)*x+16*x^2)*P(n,x) = (100*n+25+(560*n+200)*x+(960*n+560)*x^2+(512*n+640)*x^3+256*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[((100*n + 25 + (560*n + 200)*x + (960*n + 560)*x^2 + (512*n + 640)*x^3 + 256*x^4)*P[n - 1, x])/(20*n - 15 + (32*n - 12)*x + 16*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] = ((100*n + 25 + (560*n + 200)*x + (960*n + 560)*x^2 + (512*n + 640)*x^3 + 256*x^4)*v[i - 1])/(20*n - 15 + (32*n - 12)*x + 16*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 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)

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.