{"id":"n-88a6b75b97562bd9","revision":0,"first_index":0,"fingerprint":"f6-147d292a70e03bef","name":"one_plus_derivative of basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^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"],["19","4"],["345","152","16"],["6075","4140","912","64"],["104625","97200","33120","4864","256"],["1771875","2092500","972000","220800","24320","1024"],["29615625","42525000","25110000","7776000","1324800","116736","4096"],["489796875","829237500","595350000","234360000","54432000","7418880","544768","16384"],["8030390625","15673500000","13267800000","6350400000","1874880000","348364800","39567360","2490368","65536"],["130707421875","289094062500","282123000000","159213600000","57153600000","13499136000","2090188800","203489280","11206656","262144"],["2114384765625","5228296875000","5781881250000","3761640000000","1592136000000","457228800000","89994240000","11943936000","1017446400","49807360","1048576"],["34022373046875","93032929687500","115022531250000","84800925000000","41378040000000","14010796800000","3353011200000","565678080000","65691648000","4974182400","219152384","4194304"],["544934619140625","1633073906250000","2232790312500000","1840360500000000","1017611100000000","397229184000000","112086374400000","22992076800000","3394068480000","350355456000","23876075520","956301312","16777216"],["8693004638671875","28336600195312500","42459921562500000","38701698750000000","23924686500000000","10583155440000000","3442652928000000","832641638400000","149448499200000","19610173440000","1821848371200","112868720640","4143972352","67108864"],["138179849853515625","486808259765625000","793424805468750000","792585202500000000","541823782500000000","267956488800000000","98776117440000000","27541223424000000","5828491468800000","929901772800000","109816971264000","9274864435200","526720696320","17850957824","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","513","11191","240065","5082519","106472257","2211156407","45587458689","934095967255","19038574168961","386257174241463","7804847346180673","157145359253697431","3153973759669279425"],"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":[["11","4"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["225","120","16"],["60","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190470,"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-b1e4aefac6d6f7ee"]},{"id":190471,"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-b1e4aefac6d6f7ee"]},{"id":190472,"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-b1e4aefac6d6f7ee"]},{"id":190473,"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-b1e4aefac6d6f7ee"]}],"region_exact":{},"incoming_edges":[{"id":"e-b1e4aefac6d6f7ee","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-88a6b75b97562bd9","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":"5a584d86406e831670a67fe25362277611ad69054ac9709e00767be2679e997f","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-60783defa9dbe6a9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^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:           19,            4;\n  2:          345,          152,           16;\n  3:         6075,         4140,          912,           64;\n  4:       104625,        97200,        33120,         4864,         256;\n  5:      1771875,      2092500,       972000,       220800,       24320,        1024;\n  6:     29615625,     42525000,     25110000,      7776000,     1324800,      116736,       4096;\n  7:    489796875,    829237500,    595350000,    234360000,    54432000,     7418880,     544768,     16384;\n  8:   8030390625,  15673500000,  13267800000,   6350400000,  1874880000,   348364800,   39567360,   2490368,    65536;\n  9: 130707421875, 289094062500, 282123000000, 159213600000, 57153600000, 13499136000, 2090188800, 203489280, 11206656, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n+11+4*x)*P(n,x) = (60*n+225+(16*n+120)*x+16*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[((60*n + 225 + (16*n + 120)*x + 16*x^2)*P[n - 1, x])/(4*n + 11 + 4*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] = ((60*n + 225 + (16*n + 120)*x + 16*x^2)*v[i - 1])/(4*n + 11 + 4*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-a679863398df5985","steps":["basis A013611","D+1"],"edges":["e-8d978ffb07d0025e","e-b1e4aefac6d6f7ee"],"text":"A147716 \u2192 basis A013611 \u2192 D+1"},"headline":"A147716 \u2192 basis A013611 \u2192 D+1","chain":null}