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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":"2b6898d5cbbb5844e8472c8d4e739838c0cacfc2136f8e3c0516c69587682c9c","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-64714e0ad28cd8ed","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.","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:         6,            3;\n  2:        69,           78,             9;\n  3:       780,         2259,           834,             27;\n  4:      8259,        59382,         50136,           8502,             81;\n  5:     84534,      1347303,       2526024,         968670,          85506,            243;\n  6:    852873,     27549390,     104285487,       83706780,       17609343,         856518,           729;\n  7:   8556432,    527170371,    3683214738,     5706511125,     2437842540,      310633533,       8569554,        2187;\n  8:  85662735,   9672012366,  116528057478,   320287911594,   256985536920,    66004028370,    5392771314,    85708662,      6561;\n  9: 856968522, 172733246199, 3414654296808, 15545137443864, 21653348091732, 10266323644878, 1711139090136, 92793739392, 857125986, 19683;\n","recurrence_sentence":"The row polynomials satisfy (7*n-18+14*x)*P(n,x) = -(210*n-330+(294*n^2-1365*n+1839)*x+(343*n^3-2303*n^2+5432*n-5062)*x^2+(1029*n^2-4606*n+4991)*x^3+(686*n-1078)*x^4)*P(n-2,x) - ((147*n-231)*x+(343*n^2-1568*n+1911)*x^2-(343*n^2-2450*n+3983)*x^3-(1029*n-2989)*x^4-686*x^5)*d/dx P(n-2,x) + (91*n-213+(98*n^2-455*n+704)*x+(245*n-581)*x^2+98*x^3)*P(n-1,x) + ((49*n-126)*x-(49*n-224)*x^2-98*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(210*n - 330 + (294*n^2 - 1365*n + 1839)*x + (343*n^3 - 2303*n^2 + 5432*n - 5062)*x^2 + (1029*n^2 - 4606*n + 4991)*x^3 + (686*n - 1078)*x^4)*P[n - 2, x] - ((147*n - 231)*x + (343*n^2 - 1568*n + 1911)*x^2 - (343*n^2 - 2450*n + 3983)*x^3 - (1029*n - 2989)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (91*n - 213 + (98*n^2 - 455*n + 704)*x + (245*n - 581)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 126)*x - (49*n - 224)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 18 + 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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(210*n - 330 + (294*n^2 - 1365*n + 1839)*x + (343*n^3 - 2303*n^2 + 5432*n - 5062)*x^2 + (1029*n^2 - 4606*n + 4991)*x^3 + (686*n - 1078)*x^4)*v[i - 2] - ((147*n - 231)*x + (343*n^2 - 1568*n + 1911)*x^2 - (343*n^2 - 2450*n + 3983)*x^3 - (1029*n - 2989)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (91*n - 213 + (98*n^2 - 455*n + 704)*x + (245*n - 581)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 126)*x - (49*n - 224)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 18 + 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-64714e0ad28cd8ed","steps":["D+1"],"edges":["e-30e8481b3d8f7584"],"text":"A257627 \u2192 D+1"},"headline":"A257627 \u2192 D+1","chain":null}