Real-rooted atlas

A038271 → basis A013609

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(12 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121022
2144240100484
3172843203600100010648
42073669120864004800010000234256
52488321036800172800014400006000001000005153632
629859841492992031104000345600002160000072000001000000113379904
73583180820901888052254720072576000060480000030240000084000000100000002494357888
84299816962866544640836075520013934592000145152000009676800000403200000096000000010000000054875873536
9515978035238698352640128994508800250822656000313528320000261273600000145152000000518400000001080000000010000000001207269217792
10619173642245159780352001934917632000429981696000062705664000006270566400000435456000000020736000000006480000000001200000000001000000000026559922791424
1174300837068868109100646402837879193600070946979840000118244966400000137952460800000114960384000000684288000000002851200000000079200000000001320000000000100000000000584318301411328
128916100448256891610044825604086546038784001135151677440000212840939520000028378791936000002759049216000000197074944000000010264320000000003801600000000009504000000000014400000000000100000000000012855002631049216
1310699320537907211590930582732805795465291366400177083661680640003689242951680000055338644275200000614873825280000005123948544000000032024678400000000148262400000000004942080000000000112320000000000015600000000000010000000000000282810057883082752
1412839184645488641497904875307008081136514079129600270455046930432000619792815882240000103298802647040000012912350330880000001229747650560000000896690995200000000498161664000000000207567360000000000628992000000000001310400000000000016800000000000001000000000000006221821273427820544

Generated b-files: A009966 (row sums), A038314 (reversed triangle), A038336 ?

Computed in your browser ?

Technical details
Node id
n-4f44de8e265280a7 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         12,          10;
  2:        144,         240,          100;
  3:       1728,        4320,         3600,         1000;
  4:      20736,       69120,        86400,        48000,        10000;
  5:     248832,     1036800,      1728000,      1440000,       600000,       100000;
  6:    2985984,    14929920,     31104000,     34560000,     21600000,      7200000,      1000000;
  7:   35831808,   209018880,    522547200,    725760000,    604800000,    302400000,     84000000,    10000000;
  8:  429981696,  2866544640,   8360755200,  13934592000,  14515200000,   9676800000,   4032000000,   960000000,   100000000;
  9: 5159780352, 38698352640, 128994508800, 250822656000, 313528320000, 261273600000, 145152000000, 51840000000, 10800000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (12+10*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[(12 + 10*x)*P[n - 1, 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] = (12 + 10*x)*v[i - 1]); 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)

Generated with polytool c48629374597.

A038314: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         10,          12;
  2:        100,         240,         144;
  3:       1000,        3600,        4320,         1728;
  4:      10000,       48000,       86400,        69120,        20736;
  5:     100000,      600000,     1440000,      1728000,      1036800,       248832;
  6:    1000000,     7200000,    21600000,     34560000,     31104000,     14929920,      2985984;
  7:   10000000,    84000000,   302400000,    604800000,    725760000,    522547200,    209018880,     35831808;
  8:  100000000,   960000000,  4032000000,   9676800000,  14515200000,  13934592000,   8360755200,   2866544640,   429981696;
  9: 1000000000, 10800000000, 51840000000, 145152000000, 261273600000, 313528320000, 250822656000, 128994508800, 38698352640, 5159780352;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+10*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[(12 + 10*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (12 + 10*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038336: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,          10;
  2:        144,         240,          100;
  3:       1728,        4320,         3600,         1000;
  4:      20736,       69120,        86400,        48000,        10000;
  5:     248832,     1036800,      1728000,      1440000,       600000,       100000;
  6:    2985984,    14929920,     31104000,     34560000,     21600000,      7200000,      1000000;
  7:   35831808,   209018880,    522547200,    725760000,    604800000,    302400000,     84000000,    10000000;
  8:  429981696,  2866544640,   8360755200,  13934592000,  14515200000,   9676800000,   4032000000,   960000000,   100000000;
  9: 5159780352, 38698352640, 128994508800, 250822656000, 313528320000, 261273600000, 145152000000, 51840000000, 10800000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (12+10*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[(12 + 10*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (12 + 10*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e169cb33ba7b2c8c
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-e169cb33ba7b2c8c
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-e169cb33ba7b2c8c

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (6)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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