A063967 → D

nonnegative coefficients

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

Recurrence

\(\left(4 + 2\,x - 4\,n - n x + n^{2}\right)\,P_{n}(x) = \left(-30 - 16\,x - 2\,x^{2} + 17\,n + 5\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-15 - 23\,x - 9\,x^{2} - x^{3} + 5\,n + 6\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	3	2						5
4	7	10	3					20
5	15	32	21	4				72
6	30	86	87	36	5			244
7	58	208	285	184	55	6		796
8	109	470	813	716	335	78	7	2528

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-51f9b5eb10d6dd84 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-997254b322e1410f

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

OEIS matches ?

No OEIS matches.

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   1;
   3:   3,    2;
   4:   7,   10,    3;
   5:  15,   32,   21,    4;
   6:  30,   86,   87,   36,    5;
   7:  58,  208,  285,  184,   55,    6;
   8: 109,  470,  813,  716,  335,   78,   7;
   9: 201, 1012, 2115, 2364, 1515,  552, 105,   8;
  10: 365, 2102, 5151, 6984, 5700, 2850, 847, 136, 9;

%F Formula

The row polynomials satisfy (n^2-4*n+4-(n-2)*x)*P(n,x) = -(2*n^2-17*n+30-(5*n-16)*x+2*x^2)*P(n-1,x) + (5*n-15+(6*n-23)*x+(n-9)*x^2-x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 17*n + 30 - (5*n - 16)*x + 2*x^2)*P[n - 1, x] + (5*n - 15 + (6*n - 23)*x + (n - 9)*x^2 - x^3)*D[P[n - 1, x], x])/(n^2 - 4*n + 4 - (n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 17*n + 30 - (5*n - 16)*x + 2*x^2)*v[i - 1] + (5*n - 15 + (6*n - 23)*x + (n - 9)*x^2 - x^3)*deriv(v[i - 1], x))/(n^2 - 4*n + 4 - (n - 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-8a3bcadffb0d6c12
Coefficients of one sign. Every nonzero row, n >= 1: 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-8a3bcadffb0d6c12
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-8a3bcadffb0d6c12

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 derivative_polynomial, shift 0 e-8a3bcadffb0d6c12

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A063967 e-6e2f4b29fae24343
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
  - basis_map A027466 e-07199d0e633dc35b
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-12ad66072da87843
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
  - basis_map A038291 e-19f9843f81ec4d82
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-61f403043a781326
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A038255 e-20a66a93533ac8e0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-cc7a67409dceb768
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A038279 e-2d6f22aa44a34472
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-f5bf3eab46ece5d6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A027465 e-300fe6b47fa78df1
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-acc0c42734fdb8fc
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A038327 e-37edf5cceba94f47
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-4d81c1e2745a9ed0
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A007318 e-50014fa936b4943d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image reversal, shift -1 e-24564bef34b2c2e5
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): delannoy
    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-15c233e4385971d1 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-266d85ca03a298fa
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): delannoy
    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-15c233e4385971d1 (expanded above)
    
    basis_map A008288 e-e771e35dc9ee3fbd
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-33dba7e79fe9caa8
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
    
    ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
    
    basis_map A007318 e-fcd19baf8f835cf0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-1365676e8cb0fd13
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
    
    preserver_image affine, shift 0 e-3c4e27dedb9c54a8
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]
    
    theorem_check ma_wang_interval e-0178a3d1cd265fb6 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
    
    basis_map A026729 e-fa68694d112454ed
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)
  - basis_map A038231 e-5c88965e8c99a752
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-0e7024e7b55599e9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A038303 e-78cb35d9024a5ff1
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-25462190c1ff20a9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A007318 e-933c6ae38be5f965
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ preserver_image reversal, shift -1 e-24564bef34b2c2e5 (expanded above)
    
    ↵ basis_map A007318 e-fcd19baf8f835cf0 (expanded above)
    
    ↵ basis_map A026729 e-fa68694d112454ed (expanded above)
    
    ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)
  - basis_map A038243 e-a054677672f9106d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-81aeef92852c73bb
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - basis_map A038207 e-a7cf73c813009d2a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 2]
    
    ↵ preserver_image affine, shift 0 e-1365676e8cb0fd13 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-3c4e27dedb9c54a8 (expanded above)
  - basis_map A027467 e-bfdacf28446aab8d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-79b40901050ca651
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - ↵ theorem_check ma_wang_interval e-32a5b77539bbfdf5 (expanded above)
  - preserver_image affine, shift 0 e-e504746edac4d52e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ preserver_image reversal, shift -1 e-24564bef34b2c2e5 (expanded above)
    
    ↵ basis_map A007318 e-fcd19baf8f835cf0 (expanded above)
    
    ↵ basis_map A026729 e-fa68694d112454ed (expanded above)
    
    ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.