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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"1c6728725365a3c8ba87bf06e6f7837a549a44f479ae9a1fac1460169674581a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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:       17,          18,            1;\n  3:      144,         429,          146,             1;\n  4:     1167,        9746,         7788,          1170,             1;\n  5:     9358,      189361,       360008,        127430,          9362,            1;\n  6:    74893,     3317658,     13501647,      10445620,       1995843,        74898,            1;\n  7:   599180,    54632893,    429442986,     670710215,     269226328,     30611931,       599186,          1;\n  8:  4793483,   866828586,  12202212946,   35044685774,   27398897484,   6509837110,    464571926,    4793490,        1;\n  9: 38347914, 13448222673, 321272500256, 1572380898508, 2201287566324, 996016386806, 151790415952, 7011720684, 38347922, 1;\n","recurrence_sentence":"The row polynomials satisfy (7*n-20+14*x)*P(n,x) = -(56*n-104+(98*n^2-511*n+723)*x+(343*n^3-2597*n^2+6342*n-5312)*x^2+(1029*n^2-5194*n+6195)*x^3+(686*n-1274)*x^4)*P(n-2,x) - ((49*n-91)*x+(343*n^2-1666*n+2009)*x^2-(343*n^2-2646*n+4515)*x^3-(1029*n-3283)*x^4-686*x^5)*d/dx P(n-2,x) + (63*n-173+(98*n^2-511*n+786)*x+(245*n-651)*x^2+98*x^3)*P(n-1,x) + ((49*n-140)*x-(49*n-238)*x^2-98*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[(-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*P[n - 2, x] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 20 + 14*x)]];\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] = (-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*v[i - 2] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 20 + 14*x)); 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-0af3ffe51a3d7181","steps":["D+1"],"edges":["e-b7070a2f0b4841df"],"text":"A142462 \u2192 D+1"},"headline":"A142462 \u2192 D+1","chain":null}