basis-map coefficients of A030528, shifted → basis A013609 → θ+1
only rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(-1 - 4\,x - 4\,x^{2} - 2\,n x - 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | -1 | -4 | -5 | |||||||||||||
| 2 | 1 | 8 | 12 | 21 | ||||||||||||
| 3 | -1 | -12 | -36 | -32 | -81 | |||||||||||
| 4 | 1 | 16 | 72 | 128 | 80 | 297 | ||||||||||
| 5 | -1 | -20 | -120 | -320 | -400 | -192 | -1053 | |||||||||
| 6 | 1 | 24 | 180 | 640 | 1200 | 1152 | 448 | 3645 | ||||||||
| 7 | -1 | -28 | -252 | -1120 | -2800 | -4032 | -3136 | -1024 | -12393 | |||||||
| 8 | 1 | 32 | 336 | 1792 | 5600 | 10752 | 12544 | 8192 | 2304 | 41553 | ||||||
| 9 | -1 | -36 | -432 | -2688 | -10080 | -24192 | -37632 | -36864 | -20736 | -5120 | -137781 | |||||
| 10 | 1 | 40 | 540 | 3840 | 16800 | 48384 | 94080 | 122880 | 103680 | 51200 | 11264 | 452709 | ||||
| 11 | -1 | -44 | -660 | -5280 | -26400 | -88704 | -206976 | -337920 | -380160 | -281600 | -123904 | -24576 | -1476225 | |||
| 12 | 1 | 48 | 792 | 7040 | 39600 | 152064 | 413952 | 811008 | 1140480 | 1126400 | 743424 | 294912 | 53248 | 4782969 | ||
| 13 | -1 | -52 | -936 | -9152 | -57200 | -247104 | -768768 | -1757184 | -2965248 | -3660800 | -3221504 | -1916928 | -692224 | -114688 | -15411789 | |
| 14 | 1 | 56 | 1092 | 11648 | 80080 | 384384 | 1345344 | 3514368 | 6918912 | 10250240 | 11275264 | 8945664 | 4845568 | 1605632 | 245760 | 49424013 |
Compute more
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Technical details
- Node id
n-0cb3ed1742a158ee· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-7cabb77b45806852
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): no - 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 ExampleTriangle begins: 0: 1; 1: -1, -4; 2: 1, 8, 12; 3: -1, -12, -36, -32; 4: 1, 16, 72, 128, 80; 5: -1, -20, -120, -320, -400, -192; 6: 1, 24, 180, 640, 1200, 1152, 448; 7: -1, -28, -252, -1120, -2800, -4032, -3136, -1024; 8: 1, 32, 336, 1792, 5600, 10752, 12544, 8192, 2304; 9: -1, -36, -432, -2688, -10080, -24192, -37632, -36864, -20736, -5120;
%F FormulaThe row polynomials satisfy (1+2*n*x)*P(n,x) = -(1+(2*n+4)*x+(4*n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (2*n + 4)*x + (4*n + 4)*x^2)*P[n - 1, x])/(1 + 2*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(1 + (2*n + 4)*x + (4*n + 4)*x^2)*v[i - 1])/(1 + 2*n*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
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
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from
basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
Related ?
Same fingerprint ?
- A013609 → θ+1
- A156992 → basis A013611 → θ+1
- A028326 → basis A013611 → θ+1
- basis-map coefficients of A026729 → basis A013609 → θ+1
- A116666 → x ↦ x/(x+1)
- A178820 → basis A013611 → θ+1
- A178822 → basis A013611 → θ+1
- A356546 → basis A013611 → θ+1
- A116666 → reverse → x ↦ x+1
- A130123 → basis A013609 → θ+1
- A120909 → basis A013610 → θ+1
- A085880 → basis A013611 → θ+1
- A085881 → basis A013611 → θ+1
- A178821 → basis A013611 → θ+1
- basis-map coefficients of A026729, shifted → basis A013609 → θ+1
- A136572 → basis A013609 → θ+1
- A038208 → basis A013611 → θ+1
- basis-map coefficients of A112227 → basis A013609 → θ+1
- A013609 → basis A013610 → θ+1
- A196347 → basis A013611 → θ+1
- A013609 → D → θ+1
- A038221 → basis A013611 → θ+1
- A120909 → θ+1
- A134082 → basis A007318 → x ↦ x+1
References
- [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
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.