A272866 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 10 + 8\,x\)

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	P_n(1)
0																													0
1	10	8																											18
2	100	280	240	64																									684
3	750	3600	6600	5760	2400	384																							19494
4	5000	34000	96000	145600	128000	65280	17920	2048																					493848
5	31250	275000	1050000	2280000	3100000	2736000	1568000	563200	115200	10240																			11728890
6	187500	2025000	9750000	27600000	51000000	64560000	57120000	35328000	14976000	4147200	675840	49152																	267418692
7	1093750	14000000	81375000	284200000	665000000	1100400000	1324400000	1173760000	766080000	363776000	122214400	27525120	3727360	229376															5927781006
8	6250000	92500000	630000000	2618000000	7420000000	15187200000	23184000000	26873600000	23846400000	16199680000	8357888000	3216998400	894566400	169738240	19660800	1048576													128717530416
9	35156250	590625000	4612500000	22230000000	74025000000	180684000000	334656000000	480326400000	540792000000	480326400000	336568320000	185020416000	78833664000	25495142400	6045696000	990904320	100270080	4718592											2751337212642
10	195312500	3671875000	32343750000	177375000000	678750000000	1925100000000	4195800000000	7189920000000	9828720000000	10812064000000	9610304000000	6902323200000	3989606400000	1839882240000	667238400000	185991168000	38436864000	5544345600	498073600	20971520									58083785600220
11	1074218750	22343750000	219140625000	1347500000000	5826562500000	18834750000000	47239500000000	94168800000000	151608600000000	199217920000000	215035744000000	191249203200000	140153728000000	84375244800000	41462784000000	16458055680000	5193154560000	1271660544000	232849408000	29989273600	2422210560	92274688							1213951119044598
12	5859375000	133593750000	1443750000000	9838125000000	47437500000000	172177500000000	488565000000000	1111176000000000	2060388000000000	3151473600000000	4007773440000000	4258336512000000	3789167616000000	2823720345600000	1758197760000000	910275379200000	388797235200000	135405895680000	37803786240000	8252817408000	1356437913600	157789716480	11576279040	402653184					25161895922015304
13	31738281250	787109375000	9292968750000	69509375000000	369789062500000	1489166250000000	4717212500000000	12057760000000000	25315290000000000	44206448000000000	64778084800000000	80139920640000000	84023655040000000	74797259264000000	56533601280000000	36213922201600000	19586158592000000	8889945292800000	3356455731200000	1041002659840000	260569300992000	51311647129600	7649991065600	811346165760	54525952000	1744830464			517915691061481674
14	170898437500	4580078125000	58652343750000	477750000000000	2779765625000000	12300356250000000	43030487500000000	122142020000000000	286486200000000000	562602040000000000	933924992000000000	1319560569600000000	1594524422400000000	1652842723840000000	1471868697600000000	1126025019392000000	738989375488000000	414795232051200000	198182240256000000	80046994227200000	27072441876480000	7566743568384000	1716231602176000	307788093849600	41984983040000	4091627438080	253671505920	7516192768	10597351832488778868

Technical details

Node id: n-cd63d144a226e1c8 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-dc7db2765c5515ed

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
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:        0;
  1:       10,         8;
  2:      100,       280,        240,          64;
  3:      750,      3600,       6600,        5760,        2400,          384;
  4:     5000,     34000,      96000,      145600,      128000,        65280,        17920,         2048;
  5:    31250,    275000,    1050000,     2280000,     3100000,      2736000,      1568000,       563200,       115200,        10240;
  6:   187500,   2025000,    9750000,    27600000,    51000000,     64560000,     57120000,     35328000,     14976000,      4147200,       675840,        49152;
  7:  1093750,  14000000,   81375000,   284200000,   665000000,   1100400000,   1324400000,   1173760000,    766080000,    363776000,    122214400,     27525120,     3727360,      229376;
  8:  6250000,  92500000,  630000000,  2618000000,  7420000000,  15187200000,  23184000000,  26873600000,  23846400000,  16199680000,   8357888000,   3216998400,   894566400,   169738240,   19660800,   1048576;
  9: 35156250, 590625000, 4612500000, 22230000000, 74025000000, 180684000000, 334656000000, 480326400000, 540792000000, 480326400000, 336568320000, 185020416000, 78833664000, 25495142400, 6045696000, 990904320, 100270080, 4718592;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+10*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 10 + 8*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 10 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10*n*x + 4*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 10 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10*n*x + 4*n*x^2)*v[i - 1])/(n - 1)); 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_A013609 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)

Related ?

Same fingerprint ?

A272866 → D → basis A013609

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.