{"id":"n-d634ae1414b25970","revision":0,"first_index":1,"fingerprint":"f6-925e1f55d72de84d","name":"one_plus_derivative of basis_map_A013609 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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":[["1"],["10","6"],["56","80","20"],["288","672","392","56"],["1408","4608","4608","1536","144"],["6656","28160","40832","23936","5280","352"],["30720","159744","306176","266240","104832","16640","832"],["139264","860160","2058240","2406400","1428480","408576","49280","1920"],["622592","4456448","12812288","18939904","15319040","6684672","1462272","139264","4352"],["2752512","22413312","75333632","135102464","139616256","83894272","28250112","4902912","379392","9728"],["12058624","110100480","423886848","895483904","1131970560","876675072","410009600","110297088","15611904","1003520","21504"],["52428800","530579456","2303197184","5607260160","8404860928","8018984960","4869423104","1832157184","404152320","47669248","2590720","47104"],["226492416","2516582400","12163481600","33554432000","58274611200","66311946240","49912217600","24536678400","7614464000","1405747200","140574720","6553600","102400"],["973078528","11777605632","62738399232","193500020736","382489067520","506720157696","457231564800","280114495488","114187567104","29801840640","4681138176","402554880","16293888","221184"],["4160749568","54492397568","317223600128","1082063323136","2400341458944","3633961762816","3834051035136","2828739280896","1443835674624","497303945216","110885339136","15021899776","1124171776","39911424","475136"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","156","1408","12304","105216","885184","7352320","60440832","492654592","3987119104","32073351168","256663883776","2044634005504","16223245025280"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-6","24","-6"],["19","-28","4"],["-16","8","0"],["4","0","0"]]},"terms":[{"coeff":{"coeffs":[["24","108","84","-36"],["20","-54","-128","24"],["-48","-48","48","0"],["16","24","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","28","24","-16"],["0","8","0","-32","0"],["0","-8","-16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183217,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-d618d4b69b545f73"]},{"id":183218,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-d618d4b69b545f73"]},{"id":183219,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-d618d4b69b545f73"]},{"id":183220,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d618d4b69b545f73"]}],"region_exact":{},"incoming_edges":[{"id":"e-d618d4b69b545f73","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d634ae1414b25970","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":"21a4974576866597e2963adcc43c3d4b5969e3d2cf61d0c2c03e066987d67a8a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-504214918830dff1","role":"sequence","name":"basis_map_A013609 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:      10,        6;\n   3:      56,       80,       20;\n   4:     288,      672,      392,        56;\n   5:    1408,     4608,     4608,      1536,       144;\n   6:    6656,    28160,    40832,     23936,      5280,      352;\n   7:   30720,   159744,   306176,    266240,    104832,    16640,      832;\n   8:  139264,   860160,  2058240,   2406400,   1428480,   408576,    49280,    1920;\n   9:  622592,  4456448, 12812288,  18939904,  15319040,  6684672,  1462272,  139264,   4352;\n  10: 2752512, 22413312, 75333632, 135102464, 139616256, 83894272, 28250112, 4902912, 379392, 9728;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-16*n^2+19*n-6+(8*n^2-28*n+24)*x+(4*n-6)*x^2)*P(n,x) = (16*n^3-48*n^2+20*n+24+(24*n^3-48*n^2-54*n+108)*x+(48*n^2-128*n+84)*x^2+(24*n-36)*x^3)*P(n-1,x) - ((8*n^2-8*n-6)*x+(16*n^2-28)*x^2+(32*n-24)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 48*n^2 + 20*n + 24 + (24*n^3 - 48*n^2 - 54*n + 108)*x + (48*n^2 - 128*n + 84)*x^2 + (24*n - 36)*x^3)*P[n - 1, x] - ((8*n^2 - 8*n - 6)*x + (16*n^2 - 28)*x^2 + (32*n - 24)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 + 19*n - 6 + (8*n^2 - 28*n + 24)*x + (4*n - 6)*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] = ((16*n^3 - 48*n^2 + 20*n + 24 + (24*n^3 - 48*n^2 - 54*n + 108)*x + (48*n^2 - 128*n + 84)*x^2 + (24*n - 36)*x^3)*v[i - 1] - ((8*n^2 - 8*n - 6)*x + (16*n^2 - 28)*x^2 + (32*n - 24)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 16*n^2 + 19*n - 6 + (8*n^2 - 28*n + 24)*x + (4*n - 6)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013609","D+1"],"edges":["e-190e8c51191ea68a","e-d618d4b69b545f73"],"text":"A091042 \u2192 basis A013609 \u2192 D+1"},"headline":"A091042 \u2192 basis A013609 \u2192 D+1","chain":null}