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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":"c63a86c4341711ecda19064b1a12563f7f94a03b0943a10bec87d83d133ac56e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c8c19eba9d68881f","role":"sequence","name":"binomial","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]"}]},{"id":"e-fe7c4fb58c0b00e3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-14d5814e0f152cf3","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; 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(3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 8*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 3 + 8*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 8*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A141611","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2,   2;\n  2:  3,   8,   3;\n  3:  4,  18,  18,    4;\n  4:  5,  32,  54,   32,    5;\n  5:  6,  50, 120,  120,   50,    6;\n  6:  7,  72, 225,  320,  225,   72,    7;\n  7:  8,  98, 378,  700,  700,  378,   98,   8;\n  8:  9, 128, 588, 1344, 1750, 1344,  588, 128,   9;\n  9: 10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 8*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 3 + 8*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 8*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u2212D\u00b2+(\u22121+n)D+1+n"],"edges":["e-168270346f6f7c93"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n"},"headline":"A141611: Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","chain":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n"}