{"id":"n-9fc2b3f4e7fa1d35","revision":0,"first_index":0,"fingerprint":"f6-5e0e612a95dd65b2","name":"one_plus_derivative of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","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":true,"rows":[["1"],["14","9"],["115","252","81"],["800","3105","3402","729"],["5125","28800","55890","40824","6561"],["31250","230625","648000","838350","459270","59049"],["184375","1687500","6226875","11664000","11317725","4960116","531441"],["1062500","11615625","53156250","130764375","183708000","142603335","52081218","4782969"],["6015625","76500000","418162500","1275750000","2353758750","2645395200","1711240020","535692528","43046721"],["33593750","487265625","3098250000","11290387500","25833937500","38130891750","35712835200","19801491660","5423886846","387420489"],["185546875","3023437500","21926953125","92947500000","254033718750","465010875000","571963376250","459165024000","222766781175","54238868460","3486784401"],["1015625000","18369140625","149660156250","723589453125","2300450625000","5029867631250","7672679437500","8089196321250","5682167172000","2450434592925","536964797754","31381059609"],["5517578125","109687500000","991933593750","5387765625000","19536915234375","49689733500000","90537617362500","118378482750000","109204150336875","68186006064000","26464693603590","5272018014312","282429536481"],["29785156250","645556640625","6416718750000","38685410156250","157592144531250","457163816484375","968949803250000","1513271604487500","1731285310218750","1419653954379375","797776270948800","281488104692730","51402175639542","2541865828329"],["159912109375","3752929687500","40670068359375","269502187500000","1218590419921875","3971322042187500","9600440146171875","17441096458500000","23834027770678125","24237994343062500","17887639825180125","9138164558140800","2955625099273665","498205702352484","22876792454961"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","448","8036","137200","2266544","36572032","579774272","9065561344","140199960320","2148748865536","32685776012288","494046950699008","7426902521163776","111120068255580160"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-4","9"],["9","0"]]},"terms":[{"coeff":{"coeffs":[["25","90","81"],["45","81","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188143,"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-5485858af86abd16"]},{"id":188144,"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-5485858af86abd16"]},{"id":188145,"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-5485858af86abd16"]},{"id":188146,"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-5485858af86abd16"]}],"region_exact":{},"incoming_edges":[{"id":"e-5485858af86abd16","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-9fc2b3f4e7fa1d35","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":"d30b0435b02b13046255fad77f1fbe21ed8625f08ecb9635e27759ed3f604a24","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-a13c2de84911142b","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","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:       14,         9;\n  2:      115,       252,         81;\n  3:      800,      3105,       3402,         729;\n  4:     5125,     28800,      55890,       40824,        6561;\n  5:    31250,    230625,     648000,      838350,      459270,       59049;\n  6:   184375,   1687500,    6226875,    11664000,    11317725,     4960116,      531441;\n  7:  1062500,  11615625,   53156250,   130764375,   183708000,   142603335,    52081218,     4782969;\n  8:  6015625,  76500000,  418162500,  1275750000,  2353758750,  2645395200,  1711240020,   535692528,   43046721;\n  9: 33593750, 487265625, 3098250000, 11290387500, 25833937500, 38130891750, 35712835200, 19801491660, 5423886846, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (9*n-4+9*x)*P(n,x) = (45*n+25+(81*n+90)*x+81*x^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[Together[((45*n + 25 + (81*n + 90)*x + 81*x^2)*P[n - 1, x])/(9*n - 4 + 9*x)]];\nTable[CoefficientList[P[n, x], x], {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] = ((45*n + 25 + (81*n + 90)*x + 81*x^2)*v[i - 1])/(9*n - 4 + 9*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-5733d79f45a14084","steps":["basis A013610","D+1"],"edges":["e-4eef4a156684ec8f","e-5485858af86abd16"],"text":"A013620 \u2192 basis A013610 \u2192 D+1"},"headline":"A013620 \u2192 basis A013610 \u2192 D+1","chain":null}