Real-rooted atlas

A178821 → 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

\(\left(n + n x\right)\,P_{n}(x) = \left(8 + 28\,x + 20\,x^{2} + 2\,n + 7\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1102535
260300375735
328021005250437512005
4112011200420007000043750168070
54032504002520006300007875003937502117682
6134402016001260000420000078750007875000328125024706290
74224073920055440002310000057750000866250007218750025781250271769190
81267202534400221760001108800003465000006930000008662500006187500001933593752853576495
936608082368008236800048048000018018000004504500000750750000080437500005027343750139648437528852829005
101025024256256002882880001921920000840840000025225200000525525000007507500000070382812500391015625009775390625282757724249
11279552076876800960960000720720000036036000000126126000000315315000000563062500000703828125000586523437500293261718750666503906252699051004195
12745472022364160030750720002562560000014414400000057657600000016816800000003603600000000563062500000062562500000004692187500000213281250000044433593750025191142705820
1319496960633651200950476800087127040000544544000000245044800000081681600000002042040000000038288250000000531781250000005317812500000036257812500000151074218750002905273437500230595844768660
14501350401754726400285143040002851430400001960358400000980179200000036756720000000105019200000000229729500000000382882500000000478603125000000435093750000000271933593750000104589843750000186767578125002075362602917940
Technical details
Node id
n-36ada3fcb5984f7f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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:     10,      25;
  2:     60,     300,      375;
  3:    280,    2100,     5250,      4375;
  4:   1120,   11200,    42000,     70000,      43750;
  5:   4032,   50400,   252000,    630000,     787500,     393750;
  6:  13440,  201600,  1260000,   4200000,    7875000,    7875000,    3281250;
  7:  42240,  739200,  5544000,  23100000,   57750000,   86625000,   72187500,   25781250;
  8: 126720, 2534400, 22176000, 110880000,  346500000,  693000000,  866250000,  618750000,  193359375;
  9: 366080, 8236800, 82368000, 480480000, 1801800000, 4504500000, 7507500000, 8043750000, 5027343750, 1396484375;
%F Formula
The row polynomials satisfy (n+n*x)*P(n,x) = (2*n+8+(7*n+28)*x+(5*n+20)*x^2)*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[((2*n + 8 + (7*n + 28)*x + (5*n + 20)*x^2)*P[n - 1, x])/(n + n*x)]];
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] = ((2*n + 8 + (7*n + 28)*x + (5*n + 20)*x^2)*v[i - 1])/(n + n*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.