Real-rooted atlas

A147716 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + x + n\right)\,P_{n}(x) = \left(75 + 30\,x + 3\,x^{2} + 15\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118321
23151089432
354002835486278748
4911256480017010194481174960
5151875013668754860008505072902433464208
625059375273375001230187529160003827252624472968024448
7410062500526246875287043750861131251530900016074459185421871326476736
86663515625984150000063149625002296350000516678750734832006429780314928656125713241344
91076414062501799149218751328602500005683466250015500362500279006525033067440024800580106288219683495898225920
10172995117187532292421875002698723828125132860250000042625996875093002175000139503262501417176000930021753542940590499521245937664
112767921875000057088388671875532824960937502968596210937510960970625000281331579375051151196250065765823750584585100034100797511691702177147182093828557824
12441137548828125996451875000000102759099609375063938995312500026717365898437578918988500000168798947625002630632950000295946206875233834040001227628710382637525314413470494144278528
1370063022460937501720436440429687519430811562500000133586829492187506234052042968750208395454007812551297342525000094045127962500128243356312501282433563125911952756004352501790124357194159432365939388741292032
14110933118896484375294264694335937500361291652490234375272031361875000000140266170966796875523660371609375001458768178054687530778405515000004937369218031255984689961250053862209651253482001432001523375626540176939647829691249377891940270080
Technical details
Node id
n-159ea4af14a6eac8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d5fd215a74ece842

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:           18,            3;
  2:          315,          108,            9;
  3:         5400,         2835,          486,          27;
  4:        91125,        64800,        17010,        1944,          81;
  5:      1518750,      1366875,       486000,       85050,        7290,        243;
  6:     25059375,     27337500,     12301875,     2916000,      382725,      26244,       729;
  7:    410062500,    526246875,    287043750,    86113125,    15309000,    1607445,     91854,     2187;
  8:   6663515625,   9841500000,   6314962500,  2296350000,   516678750,   73483200,   6429780,   314928,    6561;
  9: 107641406250, 179914921875, 132860250000, 56834662500, 15500362500, 2790065250, 330674400, 24800580, 1062882, 19683;
%F Formula
The row polynomials satisfy (n+4+x)*P(n,x) = (15*n+75+(3*n+30)*x+3*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[((15*n + 75 + (3*n + 30)*x + 3*x^2)*P[n - 1, x])/(n + 4 + 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] = ((15*n + 75 + (3*n + 30)*x + 3*x^2)*v[i - 1])/(n + 4 + 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.