Real-rooted atlas

A013611 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(25 + 40\,x + 16\,x^{2} + 20\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
2657216153
3425780432641701
4262568006240230425618225
515625525006800041600115201024190269
6906253750006300005440002496005529640961948617
7515625253750052500005880000380800013977602580481638419663317
8289062516500000406000005600000047040000243712007454720117964865536196101729
9160156251040625002970000004872000005040000003386880001462272003833856053084162621441937102445
10878906256406250002081250000396000000048720000004032000000225792000083558400019169280023592960104857618983603961
11478515625386718750014093750000305250000004356000000042873600000295680000001419264000045957120009371648001038090244194304184799573253
122587890625229687500009281250000022550000000036630000000041817600000034298880000020275200000085155840000245104640004498391040452984832167772161788720397713
13139160156251345703125005971875000001608750000000293150000000038095200000003624192000000254791680000013178880000004920115200001274544128002126512128019629342726710886417228201725341
1474462890625779296875000376796875000011147500000000225225000000003283280000000035555520000000289935360000001783541760000082001920000002755264512000648858828800992372326408455716864268435456165221278841385
Technical details
Node id
n-5fb9511402d2ef0d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-374a4446546c6a5d

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:        9,         4;
  2:       65,        72,        16;
  3:      425,       780,       432,        64;
  4:     2625,      6800,      6240,      2304,       256;
  5:    15625,     52500,     68000,     41600,     11520,      1024;
  6:    90625,    375000,    630000,    544000,    249600,     55296,      4096;
  7:   515625,   2537500,   5250000,   5880000,   3808000,   1397760,    258048,    16384;
  8:  2890625,  16500000,  40600000,  56000000,  47040000,  24371200,   7454720,  1179648,   65536;
  9: 16015625, 104062500, 297000000, 487200000, 504000000, 338688000, 146227200, 38338560, 5308416, 262144;
%F Formula
The row polynomials satisfy (4*n+1+4*x)*P(n,x) = (20*n+25+(16*n+40)*x+16*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[((20*n + 25 + (16*n + 40)*x + 16*x^2)*P[n - 1, x])/(4*n + 1 + 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] = ((20*n + 25 + (16*n + 40)*x + 16*x^2)*v[i - 1])/(4*n + 1 + 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.