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an exact degree step keeps every row nonzero; S_n = phi(n, D) = 1 + (2) D + (1) D^2: Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a 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":"821507c3d9468f45aa92d0977aa045ee7ba5adab8f4092cba69fc79b5d483068","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1451d5fffb0086c3","kind":"fingerprint_related","node_a_id":"n-011d67b73ad91983","node_b_id":"n-9e4edf560ed848c1","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-011d67b73ad91983","other":{"id":"n-011d67b73ad91983","name":"basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. 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begins:\n   1: 1;\n   2: 0,      1;\n   3: 0,      2,       1;\n   4: 0,      6,       6,       1;\n   5: 0,     24,      36,      12,      1;\n   6: 0,    120,     240,     120,     20,      1;\n   7: 0,    720,    1800,    1200,    300,     30,     1;\n   8: 0,   5040,   15120,   12600,   4200,    630,    42,    1;\n   9: 0,  40320,  141120,  141120,  58800,  11760,  1176,   56,  1;\n  10: 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + 2*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = 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P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A105278","target":"triangle","first_row":2,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       2,        1;\n   3:       6,        6,        1;\n   4:      24,       36,       12,        1;\n   5:     120,      240,      120,       20,       1;\n   6:     720,     1800,     1200,      300,      30,      1;\n   7:    5040,    15120,    12600,     4200,     630,     42,     1;\n   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;\n   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;\n  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + 2*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A271703","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0,      1;\n  2: 0,      2,       1;\n  3: 0,      6,       6,       1;\n  4: 0,     24,      36,      12,      1;\n  5: 0,    120,     240,     120,     20,      1;\n  6: 0,    720,    1800,    1200,    300,     30,     1;\n  7: 0,   5040,   15120,   12600,   4200,    630,    42,    1;\n  8: 0,  40320,  141120,  141120,  58800,  11760,  1176,   56,  1;\n  9: 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + 2*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + 2*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] + 2*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-9e4edf560ed848c1","steps":[],"edges":[],"text":"A271703"},"headline":"A271703: Triangle read by rows: the unsigned Lah 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