{"id":"n-7a5ea6043e9c6d8c","revision":0,"first_index":0,"fingerprint":"f6-63acddefa3bc865a","name":"derivative of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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"],["20","2"],["219","438","3"],["2218","16644","6654","4"],["22217","440444","660666","88868","5"],["222216","10012494","36997296","20024988","1111080","6"],["2222215","210681258","1591617705","2122156940","526703145","13333290","7"],["22222214","4247387552","59102302542","157039613560","98503837570","12742162656","155555498","8"],["222222213","83544806880","2005273944216","9435406924296","11794258655370","4010547888432","292406824080","1777777704","9"],["2222222212","1619795773818","64167971898528","496163478774792","1085071796294040","744245218162188","149725267763232","6479183095272","19999999908","10"],["22222222211","31140564145310","1974070850391855","23833715650008360","84024541795292790","100829450154351348","41709002387514630","5264188934378280","140132538653895","222222222110","11"],["222222222210","595715163203292","59104273200845070","1073990375155116540","5771826174584137140","11091239516978648280","8080556644417791996","2147980750310233080","177312819602535210","2978575816016460","2444444444310","12"],["2222222222209","11363032545304548","1736234769400911318","46199711084031706300","363505625763774692715","1053294423641887387752","1228843494248868619044","581609001222039508344","103949349939071339175","5787449231336371060","62496678999175014","26666666666508","13"],["22222222222208","216382062805227974","50319028425021199104","1920052462129817039684","21456729171246885760640","89774286117210469408194","157291967263855183237632","119699048156280625877592","38622112508244394369152","4800131155324542599210","184503104225077730048","1298292376831367844","288888888888704","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","22","660","25520","1212200","68368080","4466714560","331813081600","27623439043200","2547494933984000","257806487319180800","28405587511895193600","3384999178500843904000","433800663952492764928000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-17"],["9"]]},"terms":[{"coeff":{"coeffs":[["80","-416","3536"],["-90","612","-7074"],["0","-162","4131"],["0","0","-729"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","72","-1296","1224"],["0","-81","2106","-2025"],["0","0","-729","729"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-178","578"],["99","-612"],["0","162"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-153","153"],["0","81","-81"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191774,"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-76ef5ddddca807ca"]},{"id":191775,"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-76ef5ddddca807ca"]},{"id":191776,"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-76ef5ddddca807ca"]},{"id":191777,"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-76ef5ddddca807ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-76ef5ddddca807ca","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-7a5ea6043e9c6d8c","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":"ebce9db6e65c84d808b5d78f0effe7d880485cc7188595ecb8601e5cde773108","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-dcd2d11144e6ea21","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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:         1;\n  2:        20,           2;\n  3:       219,         438,             3;\n  4:      2218,       16644,          6654,             4;\n  5:     22217,      440444,        660666,         88868,              5;\n  6:    222216,    10012494,      36997296,      20024988,        1111080,             6;\n  7:   2222215,   210681258,    1591617705,    2122156940,      526703145,      13333290,            7;\n  8:  22222214,  4247387552,   59102302542,  157039613560,    98503837570,   12742162656,    155555498,          8;\n  9: 222222213, 83544806880, 2005273944216, 9435406924296, 11794258655370, 4010547888432, 292406824080, 1777777704, 9;\n","recurrence_sentence":"The row polynomials satisfy (9*n-17)*P(n,x) = -(90*n-80+(162*n^2-612*n+416)*x+(729*n^3-4131*n^2+7074*n-3536)*x^2)*P(n-2,x) - ((81*n-72)*x+(729*n^2-2106*n+1296)*x^2-(729*n^2-2025*n+1224)*x^3)*d/dx P(n-2,x) + (99*n-178+(162*n^2-612*n+578)*x)*P(n-1,x) + ((81*n-153)*x-(81*n-153)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 80 + (162*n^2 - 612*n + 416)*x + (729*n^3 - 4131*n^2 + 7074*n - 3536)*x^2)*P[n - 2, x] - ((81*n - 72)*x + (729*n^2 - 2106*n + 1296)*x^2 - (729*n^2 - 2025*n + 1224)*x^3)*D[P[n - 2, x], x] + (99*n - 178 + (162*n^2 - 612*n + 578)*x)*P[n - 1, x] + ((81*n - 153)*x - (81*n - 153)*x^2)*D[P[n - 1, x], x])/(9*n - 17)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(90*n - 80 + (162*n^2 - 612*n + 416)*x + (729*n^3 - 4131*n^2 + 7074*n - 3536)*x^2)*v[i - 2] - ((81*n - 72)*x + (729*n^2 - 2106*n + 1296)*x^2 - (729*n^2 - 2025*n + 1224)*x^3)*deriv(v[i - 2], x) + (99*n - 178 + (162*n^2 - 612*n + 578)*x)*v[i - 1] + ((81*n - 153)*x - (81*n - 153)*x^2)*deriv(v[i - 1], x))/(9*n - 17)); 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-dcd2d11144e6ea21","steps":["D"],"edges":["e-76ef5ddddca807ca"],"text":"A257608 \u2192 D"},"headline":"A257608 \u2192 D","chain":null}