Real-rooted atlas

A034870 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 28\,x + 69\,x^{2} + 70\,x^{3} + 25\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
14202549
21616060010006252401
364960600020000375003750015625117649
42565120448002240007000001400000175000012500003906255764801
51024256002880001920000840000025200000525000007500000070312500390625009765625282475249
64096122880168960014080000792000003168000009240000001980000000309375000034375000002578125000117187500024414062513841287201
716384573440931840093184000640640000320320000012012000000343200000007507500000012512500000015640625000014218750000088867187500341796875006103515625678223072849
865536262144049152000573440000465920000027955200000128128000000457600000000128700000000028600000000005005000000000682500000000071093750000005468750000000292968750000097656250000015258789062533232930569601
926214411796480250675200334233600031334400000219340800000118809600000050918400000001750320000000048620000000000109395000000000198900000000000290062500000000334687500000000298828125000000199218750000000933837890625002746582031250038146972656251628413597910449
10104857652428800124518400018677760000198451200000158760960000099225600000004961280000000020155200000000067184000000000018475600000000004199000000000000787312500000000012112500000000000151406250000000001514062500000000011828613281250000695800781250000028991699218750007629394531250009536743164062579792266297612001
114194304230686720605552640010092544000011984896000001078640640000076403712000000436592640000000204652800000000079587200000000002586584000000000070543200000000000161661500000000000310887500000000000499640625000000000666187500000000000728642578125000000642919921875000000446472167968750000234985351562500000881195068359375002098083496093750023841857910156253909821048582988049
12167772161006632960289406976005305794560006963855360000696385536000005513052160000003544104960000000188280576000000008368025600000000031380096000000000099845760000000000027041560000000000006240360000000000000122578500000000000002042975000000000000028729335937500000000337992187500000000003286035156250000000025942382812500000000162139892578125000007720947265625000000263214111328125000057220458984375000059604644775390625191581231380566414401
136710886443620761601363148800002726297600000391905280000004310958080000003772088320000000269434880000000001599769600000000007998848000000000003399510400000000000123618560000000000003863080000000000000010400600000000000000024144250000000000000048288500000000000000082995859375000000000012205273437500000000001525659179687500000000160595703125000000000014052124023437500000001003723144531250000000570297241210937500000247955322265625000000774860382080078125001549720764160156250014901161193847656259387480337647754305649
14268435456187904819206341787648001374053990400021469593600000025763512320000002469003264000000019399311360000000012730798080000000007072665600000000000335951616000000000001374347520000000000004867480800000000000001497686400000000000000401166000000000000000093605400000000000000001901359687500000000000033553406250000000000000512621484375000000000006745019531250000000000075881469726562500000000722680664062500000000005748596191406250000000037490844726562500000000195264816284179687500007810592651367187500000225305557250976562500041723251342773437500037252902984619140625459986536544739960976801
Technical details
Node id
n-2d2986fb6c657dd8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b2b1f354adf1851f

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:      4,       20,        25;
  2:     16,      160,       600,       1000,         625;
  3:     64,      960,      6000,      20000,       37500,        37500,         15625;
  4:    256,     5120,     44800,     224000,      700000,      1400000,       1750000,       1250000,         390625;
  5:   1024,    25600,    288000,    1920000,     8400000,     25200000,      52500000,      75000000,       70312500,       39062500,         9765625;
  6:   4096,   122880,   1689600,   14080000,    79200000,    316800000,     924000000,    1980000000,     3093750000,     3437500000,      2578125000,      1171875000,       244140625;
  7:  16384,   573440,   9318400,   93184000,   640640000,   3203200000,   12012000000,   34320000000,    75075000000,   125125000000,    156406250000,    142187500000,     88867187500,     34179687500,      6103515625;
  8:  65536,  2621440,  49152000,  573440000,  4659200000,  27955200000,  128128000000,  457600000000,  1287000000000,  2860000000000,   5005000000000,   6825000000000,   7109375000000,   5468750000000,   2929687500000,    976562500000,   152587890625;
  9: 262144, 11796480, 250675200, 3342336000, 31334400000, 219340800000, 1188096000000, 5091840000000, 17503200000000, 48620000000000, 109395000000000, 198900000000000, 290062500000000, 334687500000000, 298828125000000, 199218750000000, 93383789062500, 27465820312500, 3814697265625;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+28*x+69*x^2+70*x^3+25*x^4)*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[((4 + 28*x + 69*x^2 + 70*x^3 + 25*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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] = ((4 + 28*x + 69*x^2 + 70*x^3 + 25*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.