{"id":"n-5548c32b67603fa1","revision":0,"first_index":0,"fingerprint":"f6-eff54fadfb80e9ba","name":"derivative of basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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":true,"rows":[["12","8"],["42","64","24"],["120","296","240","64"],["312","1072","1368","768","160"],["768","3392","5952","5184","2240","384"],["1824","9856","22080","26240","17440","6144","896"],["4224","27008","73728","111360","100480","54144","16128","2048"],["9600","70912","228480","419328","479360","349440","158592","40960","4608"],["21504","180224","669696","1447936","2007040","1849344","1132544","444416","101376","10240"],["47616","446464","1880064","4681728","7633920","8515584","6580224","3477504","1202688","245760","22528"],["104448","1083392","5099520","14376960","26972160","35352576","33030144","21995520","10229760","3164160","585728","49152"],["227328","2584576","13449216","42352640","89886720","135438336","148549632","119488512","69949440","29061120","8132608","1376256","106496"],["491520","6078464","34652160","120569856","285655040","486604800","613122048","578519040","408748032","213565440","80199680","20496384","3194880","229376"],["1056768","14123008","87539712","333545472","872734720","1658781696","2361655296","2558459904","2119163904","1335459840","630243328","215973888","50798592","7340032","491520"],["2260992","32473088","217448448","900530176","2579333120","5412126720","8593801216","10514989056","9994862592","7380172800","4198498304","1807024128","569540608","124092416","16711680","1048576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["20","130","720","3680","17920","84480","389120","1761280","7864320","34734080","152043520","660602880","2852126720","12247367680","52344913920"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["12","8"]],"recurrence":{"denominator":{"coeffs":[["1","2"],["3","2"]]},"terms":[{"coeff":{"coeffs":[["8","16","8"],["6","10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179265,"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-8d22e0cfb5198fb8"]},{"id":179266,"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-8d22e0cfb5198fb8"]},{"id":179267,"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-8d22e0cfb5198fb8"]},{"id":179268,"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-8d22e0cfb5198fb8"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d22e0cfb5198fb8","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-5548c32b67603fa1","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":"1d8ed67bef91e440a570bbadf270ea881cc1005bfc9f2deef4f42a5691678240","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-2fe6a26f73f75da6","role":"sequence","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    12,      8;\n  1:    42,     64,      24;\n  2:   120,    296,     240,      64;\n  3:   312,   1072,    1368,     768,     160;\n  4:   768,   3392,    5952,    5184,    2240,     384;\n  5:  1824,   9856,   22080,   26240,   17440,    6144,     896;\n  6:  4224,  27008,   73728,  111360,  100480,   54144,   16128,    2048;\n  7:  9600,  70912,  228480,  419328,  479360,  349440,  158592,   40960,    4608;\n  8: 21504, 180224,  669696, 1447936, 2007040, 1849344, 1132544,  444416,  101376,  10240;\n  9: 47616, 446464, 1880064, 4681728, 7633920, 8515584, 6580224, 3477504, 1202688, 245760, 22528;\n","recurrence_sentence":"The row polynomials satisfy (3*n+1+(2*n+2)*x)*P(n,x) = (6*n+8+(10*n+16)*x+(4*n+8)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 8*x.","mathematica":"P[0, x_] := 12 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 8 + (10*n + 16)*x + (4*n + 8)*x^2)*P[n - 1, x])/(3*n + 1 + (2*n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 8 + (10*n + 16)*x + (4*n + 8)*x^2)*v[i - 1])/(3*n + 1 + (2*n + 2)*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-f48bed27dbc29c5f","steps":["basis A013609","D"],"edges":["e-8866dbf598c953fc","e-8d22e0cfb5198fb8"],"text":"A391533 \u2192 basis A013609 \u2192 D"},"headline":"A391533 \u2192 basis A013609 \u2192 D","chain":null}