Real-rooted atlas

A038301 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(100 + 880\,x + 1452\,x^{2} + 220\,n x + 484\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 3520\,x + 5808\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400352058089728
18000105600348480340736802816
2160000281600013939200272588801874048062914560
3320000070400000464640000136294400018740480009894973444764729344
4640000001689600000139392000005451776000011244288000011873968128050794196992352187318272
51280000000394240000003902976000001908121600000524733440000083117776896007111187578880255422247731225563645345792
62560000000090112000000010407936000000610598912000002098933760000004432948101120005688950063104004086755963699201264340126269441829587348619264
75120000000002027520000000026763264000000018317967360000007556161536000000199482664550400003413370037862400036780803673292800227581222728499206181218395095040129478489286901760

All 15 rows

Technical details
Node id
n-7c063951a060c5eb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0c4d03a739c9a724

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:             400,             3520,               5808;
  1:            8000,           105600,             348480,              340736;
  2:          160000,          2816000,           13939200,            27258880,            18740480;
  3:         3200000,         70400000,          464640000,          1362944000,          1874048000,            989497344;
  4:        64000000,       1689600000,        13939200000,         54517760000,        112442880000,         118739681280,          50794196992;
  5:      1280000000,      39424000000,       390297600000,       1908121600000,       5247334400000,        8311777689600,        7111187578880,         2554222477312;
  6:     25600000000,     901120000000,     10407936000000,      61059891200000,     209893376000000,      443294810112000,      568895006310400,       408675596369920,       126434012626944;
  7:    512000000000,   20275200000000,    267632640000000,    1831796736000000,    7556161536000000,    19948266455040000,    34133700378624000,     36780803673292800,     22758122272849920,      6181218395095040;
  8:  10240000000000,  450560000000000,   6690816000000000,   52337049600000000,  251872051200000000,   797930658201600000,  1706685018931200000,   2452053578219520000,   2275812227284992000,   1236243679019008000,   299170970322599936;
  9: 204800000000000, 9912320000000000, 163553280000000000, 1439268864000000000, 7915978752000000000, 29257457467392000000, 75094140832972800000, 134862946802073600000, 166892896667566080000, 135986804692090880000, 65817613470971985920, 14360206575484796928;
%F Formula
The row polynomials satisfy (5+(11*n+22)*x)*P(n,x) = (100+(220*n+880)*x+(484*n+1452)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3520*x + 5808*x^2.
%t Mathematica
P[0, x_] := 400 + 3520*x + 5808*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (220*n + 880)*x + (484*n + 1452)*x^2)*P[n - 1, x])/(5 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3520*x + 5808*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (220*n + 880)*x + (484*n + 1452)*x^2)*v[i - 1])/(5 + (11*n + 22)*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

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.