derivative of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 - 4\,x + 4\,n x\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,x^{2} + 6\,x^{3} - 6\,x^{4} - 8\,n x - 10\,n x^{2} + 12\,n x^{3} + 34\,n x^{4} - 24\,n x^{5} - 8\,n^{2} x^{3} - 20\,n^{2} x^{4} + 40\,n^{2} x^{5} - 16\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x - x^{2} + 3\,x^{3} + x^{4} - 2\,x^{5} - 4\,n x^{2} - 6\,n x^{3} + 12\,n x^{4} + 6\,n x^{5} - 8\,n x^{6} - 8\,n^{2} x^{4} + 8\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(3 - 6\,x - 9\,x^{2} + 12\,x^{3} + 12\,n x + 12\,n x^{2} - 28\,n x^{3} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 4\,x^{2} - x^{3} + 4\,x^{4} + 4\,n x^{2} - 4\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x + 3\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 3 | 6 | 3 | 12 | ||||||||||||
| 2 | 7 | 32 | 48 | 28 | 5 | 120 | ||||||||||
| 3 | 15 | 122 | 345 | 460 | 305 | 90 | 7 | 1344 | ||||||||
| 4 | 31 | 412 | 1878 | 4224 | 5280 | 3756 | 1442 | 248 | 9 | 17280 | ||||||
| 5 | 63 | 1318 | 8967 | 30216 | 58870 | 70644 | 52878 | 23912 | 5931 | 630 | 11 | 253440 | ||||
| 6 | 127 | 4104 | 39924 | 189396 | 525495 | 927744 | 1082368 | 840792 | 426141 | 133080 | 22572 | 1524 | 13 | 4193280 | ||
| 7 | 255 | 12594 | 170805 | 1098708 | 4121655 | 9965742 | 16362549 | 18700056 | 14948613 | 8243310 | 3021447 | 683220 | 81861 | 3570 | 15 | 77414400 |
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Technical details
- Node id
n-e679a7daf98af7e7· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-4649fccefba70494
Observed on the stored rows ?
- Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 3, 6, 3; 2: 7, 32, 48, 28, 5; 3: 15, 122, 345, 460, 305, 90, 7; 4: 31, 412, 1878, 4224, 5280, 3756, 1442, 248, 9; 5: 63, 1318, 8967, 30216, 58870, 70644, 52878, 23912, 5931, 630, 11; 6: 127, 4104, 39924, 189396, 525495, 927744, 1082368, 840792, 426141, 133080, 22572, 1524, 13; 7: 255, 12594, 170805, 1098708, 4121655, 9965742, 16362549, 18700056, 14948613, 8243310, 3021447, 683220, 81861, 3570, 15; 8: 511, 38324, 713586, 6073912, 29804090, 94482924, 206000578, 319877120, 359861760, 294286540, 173218694, 71529816, 19740214, 3330068, 287430, 8176, 17; 9: 1023, 116030, 2937891, 32547376, 204256580, 824697528, 2300669140, 4626050096, 6871916106, 7635462340, 6360818882, 3944004240, 1786844644, 571918424, 122052660, 15668752, 986255, 18414, 19;
%F FormulaThe row polynomials satisfy (1+(4*n-4)*x)*P(n,x) = -(2+(8*n+2)*x+(10*n-4)*x^2+(8*n^2-12*n-6)*x^3+(20*n^2-34*n+6)*x^4+(16*n^3-40*n^2+24*n)*x^5)*P(n-2,x) - (x+(4*n+1)*x^2+(6*n-3)*x^3+(8*n^2-12*n-1)*x^4-(6*n-2)*x^5-(8*n^2-8*n)*x^6)*d/dx P(n-2,x) + (3+(12*n-6)*x+(12*n-9)*x^2+(16*n^2-28*n+12)*x^3)*P(n-1,x) + (x+(4*n-4)*x^2-x^3-(4*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x + 3*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*P[n - 2, x] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*D[P[n - 2, x], x] + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*P[n - 1, x] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*v[i - 2] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*deriv(v[i - 2], x) + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*v[i - 1] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 4)*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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
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Product of real-rooted factors (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
- ↑ Product of real-rooted factors (elementary) (above)
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from
Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.