Real-rooted atlas

A178820 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 36\,x + 5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1204868
2250120014402890
3250018000432003456098260
42187521000075600012096007257602923235
517500021000001008000024192000290304001393459279511992
61312500189000001134000003628800006531840006270566402508226562027555796
7937500015750000011340000004536000000108864000001567641600012541132800429981696049240640760
86445312512375000001039500000049896000000149688000000287400960000344881152000236489932800709469798401150999977765
942968750092812500008910000000049896000000017962560000004311014400000689762304000070946979840004256818790400113515167744026089332829340
10279296875067031250000723937500000463320000000019459440000000560431872000001120863744000001537184563200001383466106880007378485903360017708366168064576574255528414
11177734375004692187500005630625000000405405000000001945944000000006538371840000001569209241600000269007298560000032280875827200002582470066176000123958563176448027045504693043212474970255978412
12111083984375319921875000042229687500000337837500000000182432250000000070053984000000001961511552000000040351094784000000605266421760000006456175165440000046484461191168000202841285197824004056825703956480265093117939541255
13683593750000213281250000003071250000000002702700000000000162162000000000007005398400000000022417274880000000053801459712000000096842627481600000012912350330880000001239585631764480000811365140791296000324546056316518400599161950122803205546563698427324720
1441503906250001394531250000002175468750000000208845000000000001378377000000000006616209600000000002381835456000000000653303439360000000013719372226560000000219509955624960000002634119467499520000022988678989086720000137932073934520320005092876576043827200873064555893227520114496922060392631720
Technical details
Node id
n-addb922a253e8369 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f055f948fcf55643

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:        20,         48;
  2:       250,       1200,        1440;
  3:      2500,      18000,       43200,        34560;
  4:     21875,     210000,      756000,      1209600,        725760;
  5:    175000,    2100000,    10080000,     24192000,      29030400,      13934592;
  6:   1312500,   18900000,   113400000,    362880000,     653184000,     627056640,     250822656;
  7:   9375000,  157500000,  1134000000,   4536000000,   10886400000,   15676416000,   12541132800,    4299816960;
  8:  64453125, 1237500000, 10395000000,  49896000000,  149688000000,  287400960000,  344881152000,  236489932800,   70946979840;
  9: 429687500, 9281250000, 89100000000, 498960000000, 1796256000000, 4311014400000, 6897623040000, 7094697984000, 4256818790400, 1135151677440;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+15+(12*n+36)*x)*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[((5*n + 15 + (12*n + 36)*x)*P[n - 1, x])/(n)]];
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] = ((5*n + 15 + (12*n + 36)*x)*v[i - 1])/(n)); 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.