Real-rooted atlas

A038226 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(35 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1225 + 1680\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0122516805763481
142875882006048013824205379
2150062541160004233600193536033177612117361
352521875180075000246960000169344000580608007962624714924299
41838265625756315000012965400000118540800006096384000167215104019110297642180533641
5643392968753088286250006353046000007260624000004978713600002048385024004682022912045864714242488651484819
62251875390625123531450000002964754800000040659494400000348509952000001911826022400065548320768001284211998720110075314176146830437604321
77881563867187548640508437500013341396600000002134623456000000219561269760000015055629926400006882573680640002022633897984003467372396544026418075402248662995818654939
8275854735351562518915753281250000583686101250000001067311728000000001280774073600000001053894094848000006022251970560000023597395476480000606790169395200092463263907840063403380965376511116753300641401
99654915737304687572825650132812500024968794331250000005136437691000000000704425740480000000067624871086080000004637134017331200000227124931461120000077871405072384000017799178302259200024410301671669760152168114316902430155888444737842659
1033792205080566406252780615732343750000010486893619125000000023970042558000000000036982351375200000000040574922651648000000032459938121318400000019078494242734080000081764975326003200000249188496231628800005126163351050649600639106080130990080365203474360565761779197418239532716881
111182727177819824218751054316798513671875000433776054245625000000010906369363890000000000186966331952400000000002307698725812480000000021098959778856960000000144678581340733440000007440612754666291200000283451914463477760000077746810824268185600014539663322980024320016616758083405742080876488338465357824104972647676132430295979
1241395451223693847656253973963317474609375000017712522215029687500000048582918075510000000000091613502656676000000000012564137507201280000000001292311286454988800000000101275006938513408000000060765004163108044800000027778287617420820480000095239843259728527360000237481167608673730560004071105730434406809600429479285848025333760210357201231685877766193386212891813387462761
13144884079282928466796875149023624405297851562500071531339714542968750000002125502665803562500000000043724626267959000000000000659617219128067200000000007538482504320768000000000066461723303399424000000000455737531223310336000000002430600166524321792000000010000183542271495372800000311694032486384271360000071244350282602119168000011273831253510665011200011043753064663508582400504857282956046106624365409786560616989860302899
14507094277490249633789062555635486444644531250000000286125358858171875000000000915601148346150000000000000204048255917142000000000000033580512973792512000000000004221550202419630080000000000413539611665596416000000000031901627185631723520000000001944480133219457433600000000933350463945339568128000000349097316384750383923200000997420903956429668352000002104448500655324135424000030922508581057824030720002827200784553858197094401211657479094510655897621559177407076402401757871041
Technical details
Node id
n-54fe43a581590639 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d7d295a178a56855

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:              1225,               1680,                 576;
  1:             42875,              88200,               60480,               13824;
  2:           1500625,            4116000,             4233600,             1935360,              331776;
  3:          52521875,          180075000,           246960000,           169344000,            58060800,             7962624;
  4:        1838265625,         7563150000,         12965400000,         11854080000,          6096384000,          1672151040,           191102976;
  5:       64339296875,       308828625000,        635304600000,        726062400000,        497871360000,        204838502400,         46820229120,          4586471424;
  6:     2251875390625,     12353145000000,      29647548000000,      40659494400000,      34850995200000,      19118260224000,       6554832076800,       1284211998720,       110075314176;
  7:    78815638671875,    486405084375000,    1334139660000000,    2134623456000000,    2195612697600000,    1505562992640000,     688257368064000,     202263389798400,     34673723965440,      2641807540224;
  8:  2758547353515625,  18915753281250000,   58368610125000000,  106731172800000000,  128077407360000000,  105389409484800000,   60222519705600000,   23597395476480000,   6067901693952000,    924632639078400,    63403380965376;
  9: 96549157373046875, 728256501328125000, 2496879433125000000, 5136437691000000000, 7044257404800000000, 6762487108608000000, 4637134017331200000, 2271249314611200000, 778714050723840000, 177991783022592000, 24410301671669760, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (35+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1225 + 1680*x + 576*x^2.
%t Mathematica
P[0, x_] := 1225 + 1680*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(35 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1225 + 1680*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (35 + 24*x)*v[i - 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

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.