Real-rooted atlas

A038336 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(462 + 320\,x + 50\,x^{2} + 110\,n + 50\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 924 + 640\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
09246401001664
125168277209600100063488
26601761006720554400128000100002359296
31686643233008800251680009240000160000010000085983232
442259782410119859209902640005033600001386000001920000010000003087007744
5104309511682958184768035419507200231061600008808800000194040000022400000010000000109521666048
625442450457683447609344011832739072009445201920004621232000001409408000002587200000025600000001000000003848290697216
761460978360322289820541184037551424204800354982172160002125170432000083182176000002114112000000332640000000288000000001000000000134140418588672

All 15 rows

Technical details
Node id
n-0c568a69083f4676 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42dc7898d6edde45

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:              924,               640,               100;
  1:            25168,             27720,              9600,              1000;
  2:           660176,           1006720,            554400,            128000,             10000;
  3:         16866432,          33008800,          25168000,           9240000,           1600000,            100000;
  4:        422597824,        1011985920,         990264000,         503360000,         138600000,          19200000,          1000000;
  5:      10430951168,       29581847680,       35419507200,       23106160000,        8808800000,        1940400000,        224000000,         10000000;
  6:     254424504576,      834476093440,     1183273907200,      944520192000,      462123200000,      140940800000,      25872000000,       2560000000,       100000000;
  7:    6146097836032,    22898205411840,    37551424204800,    35498217216000,    21251704320000,     8318217600000,    2114112000000,     332640000000,     28800000000,     1000000000;
  8:  147286844570624,   614609783603200,  1144910270592000,  1251714140160000,   887455430400000,   425034086400000,  138636960000000,   30201600000000,   4158000000000,   320000000000,   10000000000;
  9: 3505909808467968, 16201552902768640, 33803538098176000, 41980043255040000, 34422138854400000, 19524019468800000, 7792291584000000, 2178580800000000, 415272000000000, 50820000000000, 3520000000000, 100000000000;
%F Formula
The row polynomials satisfy (5*n+16+5*x)*P(n,x) = (110*n+462+(50*n+320)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 924 + 640*x + 100*x^2.
%t Mathematica
P[0, x_] := 924 + 640*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((110*n + 462 + (50*n + 320)*x + 50*x^2)*P[n - 1, x])/(5*n + 16 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 924 + 640*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((110*n + 462 + (50*n + 320)*x + 50*x^2)*v[i - 1])/(5*n + 16 + 5*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.