{"id":"n-b68a63e51672af5d","revision":0,"first_index":0,"fingerprint":"f6-53addaa3581bd24f","name":"shift_x_plus_one of derivative of Even row Pascal-square read by antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["10","6"],["38","44","12"],["130","222","120","20"],["420","948","768","260","30"],["1308","3672","3972","2044","490","42"],["3970","13332","18060","12520","4620","840","56"],["11822","46218","75264","65740","32940","9324","1344","72"],["34690","154756","294480","310580","197150","76320","17304","2040","90"],["100610","504354","1098324","1358140","1045170","515130","160776","30096","2970","110"],["289032","1608456","3945216","5597912","5066460","3038448","1212904","314424","49698","4180","132"],["823800","5039280","13747800","22018600","22923100","16214100","7902440","2631200","579150","78650","5720","156"],["2332418","15555720","46723560","83381200","98179200","80062920","46144560","18818280","5341050","1015300","120120","7644","182"],["6566290","47419566","155500800","305981800","402108000","371561400","247419480","119615400","41704650","10260250","1707420","177996","10010","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","16","94","492","2426","11528","53398","242724","1087410","4815680","21126862","91963996","397682154","1710033272"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["-8","-7"],["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174168,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-f1703275d4b63aa2"]},{"id":174169,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-370246d1f7d89d4d","e-f1703275d4b63aa2"]},{"id":174170,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-370246d1f7d89d4d","e-f1703275d4b63aa2"]},{"id":174171,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-370246d1f7d89d4d"]}],"region_exact":{},"incoming_edges":[{"id":"e-370246d1f7d89d4d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-b68a63e51672af5d","premise_count":1,"theorem":"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":"7e9d00b33f7ef4b32eb3d8bb3a85a2ee6e3a7442484bf6167f07ee72a90c2bff","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-74b50e0ea94d74b6","role":"sequence","name":"basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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-f1703275d4b63aa2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b68a63e51672af5d","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"0f60c9c1d452d1965ae1c8f1976dad8427789ffbcb3f2fbfaa351b438689adea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-69a7d6f31c3b5900","role":"sequence","name":"derivative of Even row Pascal-square read by antidiagonals.","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:     0;\n  1:     2;\n  2:    10,      6;\n  3:    38,     44,     12;\n  4:   130,    222,    120,     20;\n  5:   420,    948,    768,    260,     30;\n  6:  1308,   3672,   3972,   2044,    490,    42;\n  7:  3970,  13332,  18060,  12520,   4620,   840,    56;\n  8: 11822,  46218,  75264,  65740,  32940,  9324,  1344,   72;\n  9: 34690, 154756, 294480, 310580, 197150, 76320, 17304, 2040, 90;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (8*n^2-8*n-1+(6*n^2-7*n-1)*x)*P(n-1,x) - (5*n+9*n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n - 1 + (6*n^2 - 7*n - 1)*x)*P[n - 1, x] - (5*n + 9*n*x + 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n - 1 + (6*n^2 - 7*n - 1)*x)*v[i - 1] - (5*n + 9*n*x + 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); 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-79ebafb266da3b41","steps":["basis A007318","D"],"edges":["e-1a8887aa0d003bc9","e-370246d1f7d89d4d"],"text":"A172431 \u2192 basis A007318 \u2192 D"},"headline":"A172431 \u2192 basis A007318 \u2192 D","chain":null}
