Real-rooted atlas

A038314 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(5 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(50 + 240\,x + 216\,x^{2} + 60\,n x + 72\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 480\,x + 432\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01004804321012
11000720012960691228072
21000096000259200276480103680745360
31000001200000432000069120005184000149299219208992
4100000014400000648000001382400001555200008957952020901888484441408
510000000168000000907200000241920000036288000003135283200146313216028665446412018269824
610000000019200000001209600000038707200000725760000008360755200058525286400229323571203869835264294334230784
71000000000216000000001555200000005806080000001306368000000188116992000017557585920001031956070400348285173760515978035207133863559680
8100000000002400000000001944000000000829440000000021772800000000376233984000004389396480000034398535680000174142586880005159780352000681091006464171432228926464
91000000000002640000000000237600000000001140480000000003421440000000006897623040000009656672256000009459597312000006385228185600002837879193600007492001071104089161004482564090228109879296
101000000000000288000000000002851200000000001520640000000000513216000000000011824496640000000193133445120000002270303354880000019155684556800000113515167744000004495200642662400106993205379072011590930582732896996838034280448
111000000000000031200000000000033696000000000001976832000000000074131200000000000192148070400000000358676398080000000491899060224000000498047798476800000368924295168000000194792027848704000695455834963968001506820975755264014979048753070082288190468326760448
12100000000000000336000000000000039312000000000000251596800000000000103783680000000000029889699840000000006276836966400000000983798120448000000011621115297792000000103298802647040000006817720974704640000324546056316518400010547746830286848002097066825429811201925877696823296053733910997785722880
1310000000000000003600000000000000045360000000000000031449600000000000001415232000000000000044834549760000000000104613949440000000000184462147584000000000249023899238400000000258247006617600000000204531629241139200000121704771118694400000527387341514342400001572800119072358400028888165452349440002465123451933818881256807897232419749888
1410000000000000000384000000000000000518400000000000000038707200000000000000188697600000000000000652138905600000000000167382319104000000000032793270681600000000004980477984768000000000590278872268800000000054541767797637120000003894552675798220800000210954936605736960000083882673017192448000023110532361879552000039441975230941102080314303240121561907229292334555298179121152
Technical details
Node id
n-a1b1c992dac01d83 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-572def99d14f8fb6

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          100,           480,            432;
  1:         1000,          7200,          12960,            6912;
  2:        10000,         96000,         259200,          276480,          103680;
  3:       100000,       1200000,        4320000,         6912000,         5184000,         1492992;
  4:      1000000,      14400000,       64800000,       138240000,       155520000,        89579520,        20901888;
  5:     10000000,     168000000,      907200000,      2419200000,      3628800000,      3135283200,      1463132160,       286654464;
  6:    100000000,    1920000000,    12096000000,     38707200000,     72576000000,     83607552000,     58525286400,     22932357120,      3869835264;
  7:   1000000000,   21600000000,   155520000000,    580608000000,   1306368000000,   1881169920000,   1755758592000,   1031956070400,    348285173760,     51597803520;
  8:  10000000000,  240000000000,  1944000000000,   8294400000000,  21772800000000,  37623398400000,  43893964800000,  34398535680000,  17414258688000,   5159780352000,   681091006464;
  9: 100000000000, 2640000000000, 23760000000000, 114048000000000, 342144000000000, 689762304000000, 965667225600000, 945959731200000, 638522818560000, 283787919360000, 74920010711040, 8916100448256;
%F Formula
The row polynomials satisfy (5+(6*n+12)*x)*P(n,x) = (50+(60*n+240)*x+(72*n+216)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 480*x + 432*x^2.
%t Mathematica
P[0, x_] := 100 + 480*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((50 + (60*n + 240)*x + (72*n + 216)*x^2)*P[n - 1, x])/(5 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 100 + 480*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((50 + (60*n + 240)*x + (72*n + 216)*x^2)*v[i - 1])/(5 + (6*n + 12)*x)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.