A059473 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(6 + 54\,x + 180\,x^{2} + 300\,x^{3} + 270\,x^{4} + 126\,x^{5} + 24\,x^{6} - 9\,n - 76\,n x - 252\,n x^{2} - 426\,n x^{3} - 391\,n x^{4} - 186\,n x^{5} - 36\,n x^{6} + 5\,n^{2} + 32\,n^{2} x + 90\,n^{2} x^{2} + 140\,n^{2} x^{3} + 125\,n^{2} x^{4} + 60\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 40\,x + 218\,x^{2} + 560\,x^{3} + 790\,x^{4} + 632\,x^{5} + 270\,x^{6} + 48\,x^{7} - 2\,n - 24\,n x - 112\,n x^{2} - 272\,n x^{3} - 378\,n x^{4} - 304\,n x^{5} - 132\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(2\,x + 22\,x^{2} + 92\,x^{3} + 200\,x^{4} + 250\,x^{5} + 182\,x^{6} + 72\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	18	18													40
3	8	56	120	80												264
4	16	152	508	712	356											1744
5	32	384	1752	3808	3960	1584										11520
6	64	928	5376	15944	25592	21144	7048									76096
7	128	2176	15296	57664	126080	160192	109760	31360								502656
8	256	4992	41280	189120	525456	907776	953760	558144	139536							3320320
9	512	11264	107136	578304	1953312	4284864	6111168	5471232	2793888	620864						21932544
10	1024	25088	269824	1677696	6680640	17812128	32228256	39110208	30496512	13812640	2762528					144876544
11	2048	55296	663552	4671488	21442944	67410048	148181376	227925504	240598656	166141568	67605120	12291840				956989440
12	4096	120832	1600512	12587008	65466880	237308544	614988480	1148657280	1535576256	1433891200	888394816	328154496	54692416			6321442816
13	8192	262144	3799040	33013760	191951872	788713472	2356917120	5188451328	8417995392	9947135744	8325834880	4677381632	1581796736	243353344		41756614656
14	16384	565248	8896512	84666368	544333824	2501342208	8474238976	21509165184	41117087616	58934720128	62379147648	47302674816	24306218624	7579587456	1082798208	275825459200

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Technical details

Node id: n-22966acca1036f68 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-ab8a6f908e9a31d4

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): no

OEIS matches ?

A135032: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:   2,     4;
  2:   4,    18,     18;
  3:   8,    56,    120,     80;
  4:  16,   152,    508,    712,     356;
  5:  32,   384,   1752,   3808,    3960,    1584;
  6:  64,   928,   5376,  15944,   25592,   21144,    7048;
  7: 128,  2176,  15296,  57664,  126080,  160192,  109760,   31360;
  8: 256,  4992,  41280, 189120,  525456,  907776,  953760,  558144,  139536;
  9: 512, 11264, 107136, 578304, 1953312, 4284864, 6111168, 5471232, 2793888, 620864;

%F Formula

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (5*n^2-9*n+6+(32*n^2-76*n+54)*x+(90*n^2-252*n+180)*x^2+(140*n^2-426*n+300)*x^3+(125*n^2-391*n+270)*x^4+(60*n^2-186*n+126)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) - (2*n-2+(24*n-40)*x+(112*n-218)*x^2+(272*n-560)*x^3+(378*n-790)*x^4+(304*n-632)*x^5+(132*n-270)*x^6+(24*n-48)*x^7)*d/dx P(n-1,x) + (2*x+22*x^2+92*x^3+200*x^4+250*x^5+182*x^6+72*x^7+12*x^8)*d^2/dx^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[((5*n^2 - 9*n + 6 + (32*n^2 - 76*n + 54)*x + (90*n^2 - 252*n + 180)*x^2 + (140*n^2 - 426*n + 300)*x^3 + (125*n^2 - 391*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (2*n - 2 + (24*n - 40)*x + (112*n - 218)*x^2 + (272*n - 560)*x^3 + (378*n - 790)*x^4 + (304*n - 632)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*D[P[n - 1, x], x] + (2*x + 22*x^2 + 92*x^3 + 200*x^4 + 250*x^5 + 182*x^6 + 72*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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^2 - 9*n + 6 + (32*n^2 - 76*n + 54)*x + (90*n^2 - 252*n + 180)*x^2 + (140*n^2 - 426*n + 300)*x^3 + (125*n^2 - 391*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (2*n - 2 + (24*n - 40)*x + (112*n - 218)*x^2 + (272*n - 560)*x^3 + (378*n - 790)*x^4 + (304*n - 632)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*deriv(v[i - 1], x) + (2*x + 22*x^2 + 92*x^3 + 200*x^4 + 250*x^5 + 182*x^6 + 72*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c70dee1c4800e8a5
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-c70dee1c4800e8a5
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-c70dee1c4800e8a5

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

preserver_image mobius, shift 0 e-c70dee1c4800e8a5

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-0ee228386582da48 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-c68c82cb1005adf9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-0ee228386582da48 (expanded above)

Related sequences ?

Same fingerprint (1)

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

A059473 → x ↦ x+1

References

[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