A272866 → D → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 8\,x\)

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	5	8																											13
2	50	280	480	256																									1066
3	375	3600	13200	23040	19200	6144																							65559
4	2500	34000	192000	582400	1024000	1044480	573440	131072																					3583892
5	15625	275000	2100000	9120000	24800000	43776000	50176000	36044800	14745600	2621440																			183674465
6	93750	2025000	19500000	110400000	408000000	1032960000	1827840000	2260992000	1916928000	1061683200	346030080	50331648																	9036783678
7	546875	14000000	162750000	1136800000	5320000000	17606400000	42380800000	75120640000	98058240000	93126656000	62573772800	28185722880	7633633280	939524096															432259485931
8	3125000	92500000	1260000000	10472000000	59360000000	242995200000	741888000000	1719910400000	3052339200000	4147118080000	4279238656000	3294206361600	1832071987200	695247831040	161061273600	17179869184													20254444483624
9	17578125	590625000	9225000000	88920000000	592200000000	2890944000000	10708992000000	30740889600000	69221376000000	122963558400000	172322979840000	189460905984000	161451343872000	104428103270400	49526341632000	16234976378880	3285649981440	309237645312											934236251807157
10	97656250	3671875000	64687500000	709500000000	5430000000000	30801600000000	134265600000000	460154880000000	1258076160000000	2767888384000000	4920475648000000	7067978956800000	8170713907200000	7536157655040000	5466016972800000	3047279296512000	1259499159552000	363354233241600	65283502899200	5497558138880									42559651471214930
11	537109375	22343750000	438281250000	5390000000000	46612500000000	301356000000000	1511664000000000	6026803200000000	19405900800000000	50999787520000000	110098300928000000	195839184076800000	287034834944000000	345601002700800000	339663126528000000	269648784261120000	170169288622080000	83339545411584000	30520037605376000	7861508138598400	1269935930081280	96757023244288							1919440281351793343
12	2929687500	133593750000	2887500000000	39352500000000	379500000000000	2754840000000000	15634080000000000	71115264000000000	263729664000000000	806777241600000000	2051980001280000000	4360536588288000000	7760215277568000000	11565958535577600000	14403156049920000000	14913951812812800000	12740107803033600000	8873960779284480000	4955017870049280000	2163426566602752000	711164120845516800	165454509747732480	24277216741294080	1688849860263936					85851328947734756796
13	15869140625	787109375000	18585937500000	278037500000000	2958312500000000	23826660000000000	150950800000000000	771696640000000000	3240357120000000000	11316850688000000000	33166379417600000000	82063278735360000000	172080445521920000000	306369573945344000000	463123261685760000000	593328901351014400000	641799244742656000000	582611454708940800000	439937365599846400000	272892601261096960000	136613357678493696000	53804161700567449600	16043194063205171200	3403032468431831040	457396837154816000	29273397577908224			3813229860761885447689
14	85449218750	4580078125000	117304687500000	1911000000000000	22238125000000000	196805700000000000	1376975600000000000	7817089280000000000	36670233600000000000	144026122240000000000	478169595904000000000	1351230023270400000000	3265586017075200000000	6770043796848640000000	12057548370739200000000	18448793917718528000000	24215203855990784000000	27184020327707443200000	25976142594834432000000	20983839254695116800000	14193756406535946240000	7934305703961821184000	3599198536966602752000	1290956833185752678400	352195564609208320000	68646117320194785280	8511803295730237440	504403158265495552	168368764621332480536422

Technical details

Node id: n-877f873b9e80c2e2 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-dc1f92b0a0065af1

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:        5,         8;
  2:       50,       280,        480,         256;
  3:      375,      3600,      13200,       23040,        19200,          6144;
  4:     2500,     34000,     192000,      582400,      1024000,       1044480,         573440,         131072;
  5:    15625,    275000,    2100000,     9120000,     24800000,      43776000,       50176000,       36044800,       14745600,         2621440;
  6:    93750,   2025000,   19500000,   110400000,    408000000,    1032960000,     1827840000,     2260992000,     1916928000,      1061683200,       346030080,        50331648;
  7:   546875,  14000000,  162750000,  1136800000,   5320000000,   17606400000,    42380800000,    75120640000,    98058240000,     93126656000,     62573772800,     28185722880,      7633633280,       939524096;
  8:  3125000,  92500000, 1260000000, 10472000000,  59360000000,  242995200000,   741888000000,  1719910400000,  3052339200000,   4147118080000,   4279238656000,   3294206361600,   1832071987200,    695247831040,   161061273600,    17179869184;
  9: 17578125, 590625000, 9225000000, 88920000000, 592200000000, 2890944000000, 10708992000000, 30740889600000, 69221376000000, 122963558400000, 172322979840000, 189460905984000, 161451343872000, 104428103270400, 49526341632000, 16234976378880, 3285649981440, 309237645312;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+20*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 8*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 20*n*x + 16*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] = 5 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 20*n*x + 16*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from derivative 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 0<=k<=…?
  - Laguerre’s theorem on differential operators [SP14] 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 → basis A013611 → D

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.