{"id":"n-96bb9a675a290cdc","revision":0,"first_index":0,"fingerprint":"f6-8e3b879cee2d5239","name":"one_plus_derivative of 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(n) = 5*n + 3.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","3"],["57","66","9"],["528","1503","582","27"],["4575","30546","26508","4818","81"],["38058","538275","1000920","401490","39030","243"],["310053","8554794","31147575","25499220","5669355","313698","729"],["2500836","127011903","833172462","1284859725","568255080","77248521","2513958","2187"],["20078859","1803655434","19997833266","53651518446","43664112300","11719429974","1032297654","20124786","6561"],["160880190","24872935923","444552153504","1944862866564","2701196241948","1314591605250","230557635888","13642921764","161037654","19683"],["1287887889","336378812850","9355663875045","63497677599240","141164567492130","115863622132188","36547685761650","4403070063720","179135951565","1288419330","59049"],["10305937464","4490120121087","189239406543750","1917868841707695","6486389581493880","8431507927752798","4451688564229284","962549250448950","82492331077920","2342964783555","10307708934","177147"],["82456888503","59412453640674","3718202621157816","54628137701829450","270117759419832555","528336069110218116","438972419128802328","158150870947898532","24415269157795725","1526162527256730","30572018569488","82462734354","531441"],["659685931602","781514586443427","71500839440388408","1487938257206924358","10425741686667583650","29460982900697333469","36598728403588009824","20670333834499110036","5306464867729904406","603093403267070925","27998172624695592","398343663100422","659705063478","1594323"],["5277587895165","10239511536989274","1353038909268209427","39151843135184476572","379200844881286255509","1498803212457138258990","2671298467217675216163","2247552146866666596264","902628408023398498959","170661381860421347478","14618013872839839825","510712667820352764","5185725557397087","5277650073762","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","132","2640","66528","2018016","71495424","2895564672","131909057280","6674598298368","371350378054656","22528589601982464","1479955040007155712","104653963543363153920","7925793505684036190208"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["-12","10"],["5","0"]]},"terms":[{"coeff":{"coeffs":[["168","-849","1808","-1585","350"],["-120","645","-1810","1550","-250"],["0","-150","775","-375","0"],["0","0","-125","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","105","-675","1345","-1025","250"],["0","-75","550","-850","375","0"],["0","0","-125","125","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-117","338","-275","50"],["55","-215","125","0"],["0","50","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-60","110","-50"],["0","25","-25","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191007,"position":0,"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-de9ff7b92add189c"]},{"id":191008,"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-de9ff7b92add189c"]},{"id":191009,"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-de9ff7b92add189c"]},{"id":191010,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-de9ff7b92add189c"]}],"region_exact":{},"incoming_edges":[{"id":"e-de9ff7b92add189c","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-96bb9a675a290cdc","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":"bf83dc7d97c710f7585de8497eb7e9c7ddfe6f1ca0b988ffadf1d4b1b4b3c128","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-4155b181fd90d601","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(n) = 5*n + 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:        57,          66,            9;\n  3:       528,        1503,          582,            27;\n  4:      4575,       30546,        26508,          4818,            81;\n  5:     38058,      538275,      1000920,        401490,         39030,           243;\n  6:    310053,     8554794,     31147575,      25499220,       5669355,        313698,          729;\n  7:   2500836,   127011903,    833172462,    1284859725,     568255080,      77248521,      2513958,        2187;\n  8:  20078859,  1803655434,  19997833266,   53651518446,   43664112300,   11719429974,   1032297654,    20124786,      6561;\n  9: 160880190, 24872935923, 444552153504, 1944862866564, 2701196241948, 1314591605250, 230557635888, 13642921764, 161037654, 19683;\n","recurrence_sentence":"The row polynomials satisfy (5*n-12+10*x)*P(n,x) = -(120*n-168+(150*n^2-645*n+849)*x+(125*n^3-775*n^2+1810*n-1808)*x^2+(375*n^2-1550*n+1585)*x^3+(250*n-350)*x^4)*P(n-2,x) - ((75*n-105)*x+(125*n^2-550*n+675)*x^2-(125*n^2-850*n+1345)*x^3-(375*n-1025)*x^4-250*x^5)*d/dx P(n-2,x) + (55*n-117+(50*n^2-215*n+338)*x+(125*n-275)*x^2+50*x^3)*P(n-1,x) + ((25*n-60)*x-(25*n-110)*x^2-50*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[(-(120*n - 168 + (150*n^2 - 645*n + 849)*x + (125*n^3 - 775*n^2 + 1810*n - 1808)*x^2 + (375*n^2 - 1550*n + 1585)*x^3 + (250*n - 350)*x^4)*P[n - 2, x] - ((75*n - 105)*x + (125*n^2 - 550*n + 675)*x^2 - (125*n^2 - 850*n + 1345)*x^3 - (375*n - 1025)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (55*n - 117 + (50*n^2 - 215*n + 338)*x + (125*n - 275)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 60)*x - (25*n - 110)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 12 + 10*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] = (-(120*n - 168 + (150*n^2 - 645*n + 849)*x + (125*n^3 - 775*n^2 + 1810*n - 1808)*x^2 + (375*n^2 - 1550*n + 1585)*x^3 + (250*n - 350)*x^4)*v[i - 2] - ((75*n - 105)*x + (125*n^2 - 550*n + 675)*x^2 - (125*n^2 - 850*n + 1345)*x^3 - (375*n - 1025)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (55*n - 117 + (50*n^2 - 215*n + 338)*x + (125*n - 275)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 60)*x - (25*n - 110)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 12 + 10*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-4155b181fd90d601","steps":["D+1"],"edges":["e-de9ff7b92add189c"],"text":"A257623 \u2192 D+1"},"headline":"A257623 \u2192 D+1","chain":null}