Real-rooted atlas

A178822 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(25 + 45\,x + 5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1305484
2525189017014116
37000378006804040824153664
478750567000153090018370808266864840416
5787500708750025515000459270004133430014880348135531648
672187507796250035083125084199500011366932508184191402455257423478645632
761875000779625000420997500012629925000227338650002455257420014731544520378811144883487495168
8502734375723937500045608062500164189025000369425306250531972441000478775196900246227244120554011299271899340515072
9391015625063344531250456080625000191553862500051719542875009309517717500111714212610008617953544200387807909489077561581897841363415661568
1029326171875527871093750427575585937520523628125000646494285937501396427657625002094641486437502154488386050001454279660583755817118642335010470813556203868631728892928
112132812500004222968750000380067187500002052362812500007388506125000001861903543500000335142637830000043089767721000003878079094890000232684745693400083766508449624013707246837211217688500661092352
1215107421875003263203125000032305710937500019383426562500007850287757812500226088287425000004747854035925000073252605125700000824091807664125006592734461313000035600766091090200116511598116295201747673971744428350821929778331648
1310458984375000244740234375000264319453125000017445083906250000785028775781250002543493233531250006104383760475000001098789076885500000148336525379542500014833652537954250001068022982732706000524302191523328400157290657456998520217787064171228726800548177241505792
147097167968750017884863281250002092529003906250015066208828125000074577733699218750026847984131718750007248955715564062500149121374720175000002348661651842756250028183939822113075000253655458399017675001660290273157206600074713062292074297002068977109626672840266011342666286508129210415367588610048
Technical details
Node id
n-ee54987254c3cd08 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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,          54;
  2:        525,        1890,         1701;
  3:       7000,       37800,        68040,         40824;
  4:      78750,      567000,      1530900,       1837080,        826686;
  5:     787500,     7087500,     25515000,      45927000,      41334300,      14880348;
  6:    7218750,    77962500,    350831250,     841995000,    1136693250,     818419140,      245525742;
  7:   61875000,   779625000,   4209975000,   12629925000,   22733865000,   24552574200,    14731544520,    3788111448;
  8:  502734375,  7239375000,  45608062500,  164189025000,  369425306250,  531972441000,   478775196900,  246227244120,   55401129927;
  9: 3910156250, 63344531250, 456080625000, 1915538625000, 5171954287500, 9309517717500, 11171421261000, 8617953544200, 3878079094890, 775615818978;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+25+(9*n+45)*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 + (9*n + 45)*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 + (9*n + 45)*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.