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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"61f211ef2b4a288d0e7e7e3eb3000db8eb283f6c81d80654a4eebddd117fe122","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c826b75746f15e6c","role":"sequence","name":"reversal of Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2,    1;\n  2:  4,    7,     2;\n  3:  7,   26,    25,      5;\n  4: 11,   72,   142,     97,     15;\n  5: 16,  165,   555,    764,    411,      52;\n  6: 22,  331,  1715,   3944,   4226,    1892,     203;\n  7: 29,  602,  4487,  15624,  27566,   24380,    9402,    877;\n  8: 37, 1016, 10360,  51254, 135121,  194648,  147386,  50127,   4140;\n  9: 46, 1617, 21702, 145656, 539826, 1149861, 1406077, 934520, 285151, 21147;\n","recurrence_sentence":"The row polynomials satisfy (24*n-24)*P(n,x) = (14*n^2-73*n+35-(27*n^2-64*n+67)*x-(24*n^3+3*n^2-219*n+9)*x^2-(48*n^3-84*n^2-271*n+547)*x^3-(24*n^3-80*n^2+217*n-236)*x^4-(16*n^2+6*n-76)*x^5)*P(n-2,x) + (7*n-7+(7*n^2-42*n+47)*x+(7*n^2-44*n+46)*x^2+(176*n-368)*x^3+(32*n^2+103*n-237)*x^4+(32*n^2-102*n+361)*x^5+(16*n+38)*x^6)*d/dx P(n-2,x) - ((7*n-7)*x^2+(7*n+5)*x^3-(24*n-69)*x^4-(16*n+5)*x^5+(8*n-62)*x^6)*d^2/dx^2 P(n-2,x) + (55*n-31+(48*n^2-124*n+130)*x+(48*n^2-112*n-77)*x^2+(16*n+38)*x^3)*P(n-1,x) - (7*n-7+12*x+33*x^2+(32*n-86)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n^2 - 73*n + 35 - (27*n^2 - 64*n + 67)*x - (24*n^3 + 3*n^2 - 219*n + 9)*x^2 - (48*n^3 - 84*n^2 - 271*n + 547)*x^3 - (24*n^3 - 80*n^2 + 217*n - 236)*x^4 - (16*n^2 + 6*n - 76)*x^5)*P[n - 2, x] + (7*n - 7 + (7*n^2 - 42*n + 47)*x + (7*n^2 - 44*n + 46)*x^2 + (176*n - 368)*x^3 + (32*n^2 + 103*n - 237)*x^4 + (32*n^2 - 102*n + 361)*x^5 + (16*n + 38)*x^6)*D[P[n - 2, x], x] - ((7*n - 7)*x^2 + (7*n + 5)*x^3 - (24*n - 69)*x^4 - (16*n + 5)*x^5 + (8*n - 62)*x^6)*D[P[n - 2, x], {x, 2}] + (55*n - 31 + (48*n^2 - 124*n + 130)*x + (48*n^2 - 112*n - 77)*x^2 + (16*n + 38)*x^3)*P[n - 1, x] - (7*n - 7 + 12*x + 33*x^2 + (32*n - 86)*x^3)*D[P[n - 1, x], x])/(24*n - 24)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((14*n^2 - 73*n + 35 - (27*n^2 - 64*n + 67)*x - (24*n^3 + 3*n^2 - 219*n + 9)*x^2 - (48*n^3 - 84*n^2 - 271*n + 547)*x^3 - (24*n^3 - 80*n^2 + 217*n - 236)*x^4 - (16*n^2 + 6*n - 76)*x^5)*v[i - 2] + (7*n - 7 + (7*n^2 - 42*n + 47)*x + (7*n^2 - 44*n + 46)*x^2 + (176*n - 368)*x^3 + (32*n^2 + 103*n - 237)*x^4 + (32*n^2 - 102*n + 361)*x^5 + (16*n + 38)*x^6)*deriv(v[i - 2], x) - ((7*n - 7)*x^2 + (7*n + 5)*x^3 - (24*n - 69)*x^4 - (16*n + 5)*x^5 + (8*n - 62)*x^6)*deriv(deriv(v[i - 2], x), x) + (55*n - 31 + (48*n^2 - 124*n + 130)*x + (48*n^2 - 112*n - 77)*x^2 + (16*n + 38)*x^3)*v[i - 1] - (7*n - 7 + 12*x + 33*x^2 + (32*n - 86)*x^3)*deriv(v[i - 1], x))/(24*n - 24)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c15181c2ebe13c1d","steps":["reverse","D+1"],"edges":["e-130954a7e277f383","e-d5f711f7b5724947"],"text":"A049020 \u2192 reverse \u2192 D+1"},"headline":"A049020 \u2192 reverse \u2192 D+1","chain":null}