Real-rooted atlas

A178822 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(25 + 60\,x + 5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13072102
2525252030246069
370005040012096096768275128
47875075600027216004354560261273610523646
578750094500004536000010886400013063680062705664357803964
67218750103950000623700000199584000035925120003448811520137952460811151556878
7618750001039500000748440000029937600000718502400001034643456008277147648028378791936324988229016
850273437596525000008108100000038918880000011675664000002241727488000269007298560018446214758405533864427528977799826567
93910156250844593750008108100000004540536000000163459296000003923023104000062768369664000645617516544003873705099264010329880264704237412928746994
10293261718757038281250007601343750000486486000000002043241200000005884534656000001176906931200000161404379136000014526394122240007747410198528001859378447646726054029683048347
112132812500005630625000000675675000000004864860000000002335132800000000784604620800000018830510899200000322808758272000003873705099264000030989640794112000148750275811737603245460563165184149699643071740944
1215107421875004350937500000057432375000000045945900000000002481078600000000095273418240000000266765571072000000548774889062400000823162333593600000878039822499840000632188672199884800275864147869040640551728295738081283605266403977761068
131045898437500032632031250000046990125000000004135131000000000024810786000000000010718259552000000003429843056640000000823162333593600000014816922004684800000197558960062464000001896566016599654400012413886654106828800496555466164273152091671778368788889684862424585938068216
14709716796875002384648437500000372005156250000003571249500000000002357024670000000000113137184160000000004072938629760000000011171488813056000000023460126507417600000037536202411868160000045043442894241792000039310641071338291200023586384642802974720087088189450349445120149294039057741905921957897367232714002412
Technical details
Node id
n-c913bfd1bccb393f · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f055f948fcf55643

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         30,          72;
  2:        525,        2520,         3024;
  3:       7000,       50400,       120960,         96768;
  4:      78750,      756000,      2721600,       4354560,        2612736;
  5:     787500,     9450000,     45360000,     108864000,      130636800,       62705664;
  6:    7218750,   103950000,    623700000,    1995840000,     3592512000,     3448811520,     1379524608;
  7:   61875000,  1039500000,   7484400000,   29937600000,    71850240000,   103464345600,    82771476480,    28378791936;
  8:  502734375,  9652500000,  81081000000,  389188800000,  1167566400000,  2241727488000,  2690072985600,  1844621475840,   553386442752;
  9: 3910156250, 84459375000, 810810000000, 4540536000000, 16345929600000, 39230231040000, 62768369664000, 64561751654400, 38737050992640, 10329880264704;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+25+(12*n+60)*x)*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[((5*n + 25 + (12*n + 60)*x)*P[n - 1, x])/(n)]];
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] = ((5*n + 25 + (12*n + 60)*x)*v[i - 1])/(n)); 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.