{"id":"n-94eba4b323debfb3","revision":0,"first_index":0,"fingerprint":"f6-b7535a41ad3fce2e","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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":[[],["4"],["104","32"],["2028","1248","192"],["35152","32448","9984","1024"],["571220","703040","324480","66560","5120"],["8911032","13709280","8436480","2595840","399360","24576"],["135150652","249508896","191929920","78740480","18170880","2236416","114688"],["2007952544","4324820864","3992142336","2047252480","629923840","116293632","11927552","524288"],["29366305956","72286291584","77846775552","47905708032","18425272320","4535451648","697761792","61341696","2359296"],["424179974920","1174652238240","1445725831680","1037957007360","479057080320","147402178560","30236344320","3987210240","306708480","10485760"],["6065773641356","18663918896480","25842349241280","21203978864640","11417527080960","4215702306816","1080949309440","190057021440","21929656320","1499463680","46137344"],["86023698913776","291157134785088","447934053515520","413477587860480","254447746375680","109608259977216","33725618454528","7412223836160","1140342128640","116958167040","7197425664","201326592"],["1211500426369012","4473232343516352","7570085504412288","7764190260935680","5375208642186240","2646256562307072","949938253135872","250533165662208","48179454935040","6588643409920","608182468608","34024194048","872415232"],["16961005969166168","67844023876664672","125250505618457856","141308262749029376","108698663653099520","60202336792485888","24698394581532672","7599506025086976","1753732159635456","299783275151360","36896403095552","3096201658368","158779572224","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","136","3468","78608","1670420","34076568","675851932","13130837536","251127267876","4743515059880","88703731619756","1645051022766384","30296356335947572","554656369842732472"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["13","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189484,"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-14dcc6c26d9a99dc"]},{"id":189485,"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-14dcc6c26d9a99dc"]},{"id":189486,"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-14dcc6c26d9a99dc"]},{"id":189487,"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-14dcc6c26d9a99dc"]}],"region_exact":{},"incoming_edges":[{"id":"e-14dcc6c26d9a99dc","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-94eba4b323debfb3","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":"cd6304e6da96f5082e18444a04a22d05506fd36b25d5f8a0f99e9a58bb12b255","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-0088aa50f9dba3f3","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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":[{"id":"l-11751965cbb33eea","kind":"fingerprint_related","node_a_id":"n-94eba4b323debfb3","node_b_id":"n-ccae868e92935003","fingerprint":"f6-b7535a41ad3fce2e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7535a41ad3fce2e"},"other_id":"n-ccae868e92935003","other":{"id":"n-ccae868e92935003","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b7535a41ad3fce2e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","13"],["16","104","169"]],"card_rows":[["1"],["4","13"],["16","104","169"],["64","624","2028","2197"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013611","reverse"],"edges":["e-950a320de9db6fb9","e-3f8c1cda1cf2f126"],"text":"A038327 \u2192 basis A013611 \u2192 reverse"},"headline":"A038327 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-3f1291900ee74779","kind":"fingerprint_related","node_a_id":"n-0088aa50f9dba3f3","node_b_id":"n-94eba4b323debfb3","fingerprint":"f6-b7535a41ad3fce2e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7535a41ad3fce2e"},"other_id":"n-0088aa50f9dba3f3","other":{"id":"n-0088aa50f9dba3f3","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b7535a41ad3fce2e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","4"],["169","104","16"]],"card_rows":[["1"],["13","4"],["169","104","16"],["2197","2028","624","64"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013611"],"edges":["e-950a320de9db6fb9"],"text":"A038327 \u2192 basis A013611"},"headline":"A038327 \u2192 basis A013611","chain":null}},{"id":"l-965eab61816cf719","kind":"fingerprint_related","node_a_id":"n-fde50d1ca3577303","node_b_id":"n-94eba4b323debfb3","fingerprint":"f6-b7535a41ad3fce2e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7535a41ad3fce2e"},"other_id":"n-fde50d1ca3577303","other":{"id":"n-fde50d1ca3577303","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-b7535a41ad3fce2e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","13"],["16","104","169"]],"card_rows":[["1"],["4","13"],["16","104","169"],["64","624","2028","2197"]],"row_sums_preview":["1","17","289","4913","83521","1419857","24137569","410338673"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-39440d1ff583ab52","e-31e08ec7013f32c0"],"text":"A013610 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A013610 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:           4;\n  2:         104,          32;\n  3:        2028,        1248,         192;\n  4:       35152,       32448,        9984,        1024;\n  5:      571220,      703040,      324480,       66560,        5120;\n  6:     8911032,    13709280,     8436480,     2595840,      399360,      24576;\n  7:   135150652,   249508896,   191929920,    78740480,    18170880,    2236416,    114688;\n  8:  2007952544,  4324820864,  3992142336,  2047252480,   629923840,  116293632,  11927552,   524288;\n  9: 29366305956, 72286291584, 77846775552, 47905708032, 18425272320, 4535451648, 697761792, 61341696, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (13*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((13*n + 4*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((13*n + 4*n*x)*v[i - 1])/(n - 1)); 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-81e6fb4f9f226d77","steps":["basis A013611","D"],"edges":["e-950a320de9db6fb9","e-14dcc6c26d9a99dc"],"text":"A038327 \u2192 basis A013611 \u2192 D"},"headline":"A038327 \u2192 basis A013611 \u2192 D","chain":null}
