{"id":"n-8fb26d3bf722c8bb","revision":0,"first_index":1,"fingerprint":"f6-6a81dd80276de4b8","name":"derivative of basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["6"],["54","108"],["396","1620","1620"],["2700","16740","34020","22680"],["17730","147420","453600","612360","306180"],["113778","1186920","4898880","10001880","10103940","4041576"],["718872","9022104","46743480","127983240","195342840","157621464","52540488"],["4491288","65885400","410893560","1412306280","2889726840","3520212696","2364321960","675520560"],["27823230","467133480","3406898880","14098364280","36209152980","59108049000","59896156320","34451548560","8612887140"],["171223470","3237733620","27037501920","130874463720","404693108820","829088900640","1125417252960","976127209200","490934566980","109096570440"],["1048102308","22041290700","207384116940","1149673444920","4158754760160","10257374391264","17470763069760","20291962101840","15382616432040","6873083937720","1374616787544"],["6387670692","147875852124","1547886560220","9670543842960","40068426958560","115610720281056","237045028252032","345399742013040","350527280823720","235975881861720","94848558340536","17245192425552"],["38787497658","980191894500","11298073026240","78544001455920","366804148871160","1212310183148064","2907738770096160","5099828281788240","6491627779176540","5848994430999720","3541012844713344","1293389431916400","215564905319400"],["234796916970","6431450472720","80943430014720","619766487259920","3221100485923320","12000065143706640","32969419973249760","67638782707095600","103622049987035580","117085275907039440","94848558340536000","52166707087294800","17460757330871400","2686270358595600"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","6","162","3636","76140","1537290","30346974","589972488","11343358584","216278013870","4096678531770","77189318435196","1448093524882212","27068131829903346","504406363486012470"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-6"],["3"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["9","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","18","18"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188924,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1296c7983fa47a9a"]},{"id":188925,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1296c7983fa47a9a"]},{"id":188926,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1296c7983fa47a9a"]},{"id":188927,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-1296c7983fa47a9a"]}],"region_exact":{},"incoming_edges":[{"id":"e-1296c7983fa47a9a","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-8fb26d3bf722c8bb","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":"329dc88df8846682c9319154240075fd74d116931fce1ce11a09e73202cdd02b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ecc015942a4103af","role":"sequence","name":"basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-37e4d94a471c8442","kind":"fingerprint_related","node_a_id":"n-a84321342ff00533","node_b_id":"n-8fb26d3bf722c8bb","fingerprint":"f6-6a81dd80276de4b8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a81dd80276de4b8"},"other_id":"n-a84321342ff00533","other":{"id":"n-a84321342ff00533","name":"reversal of basis_map_A013610 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-6a81dd80276de4b8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["45","45","11"]],"card_rows":[["1"],["6","3"],["45","45","11"],["378","567","279","45"]],"row_sums_preview":["1","9","101","1269","17081","240849","3511741","52515549"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["basis A013610","reverse"],"edges":["e-13974c051c6d411c","e-087f697a1058c307"],"text":"A033282 \u2192 basis A013610 \u2192 reverse"},"headline":"A033282 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-45bfb1847f841efd","kind":"fingerprint_related","node_a_id":"n-bc0842751fa5b570","node_b_id":"n-8fb26d3bf722c8bb","fingerprint":"f6-6a81dd80276de4b8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a81dd80276de4b8"},"other_id":"n-bc0842751fa5b570","other":{"id":"n-bc0842751fa5b570","name":"basis_map_A013610 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-6a81dd80276de4b8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["11","45","45"]],"card_rows":[["1"],["3","6"],["11","45","45"],["45","279","567","378"]],"row_sums_preview":["1","9","101","1269","17081","240849","3511741","52515549"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["basis A013610"],"edges":["e-13974c051c6d411c"],"text":"A033282 \u2192 basis A013610"},"headline":"A033282 \u2192 basis A013610","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        0;\n   2:        6;\n   3:       54,       108;\n   4:      396,      1620,       1620;\n   5:     2700,     16740,      34020,       22680;\n   6:    17730,    147420,     453600,      612360,      306180;\n   7:   113778,   1186920,    4898880,    10001880,    10103940,     4041576;\n   8:   718872,   9022104,   46743480,   127983240,   195342840,   157621464,    52540488;\n   9:  4491288,  65885400,  410893560,  1412306280,  2889726840,  3520212696,  2364321960,   675520560;\n  10: 27823230, 467133480, 3406898880, 14098364280, 36209152980, 59108049000, 59896156320, 34451548560, 8612887140;\n","recurrence_sentence":"The row polynomials satisfy (3*n-6)*P(n,x) = (9*n+18*n*x)*P(n-1,x) + (4+18*x+18*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n + 18*n*x)*P[n - 1, x] + (4 + 18*x + 18*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = ((9*n + 18*n*x)*v[i - 1] + (4 + 18*x + 18*x^2)*deriv(v[i - 1], x))/(3*n - 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4689fb1386d440ad","steps":["basis A013610","D"],"edges":["e-50c3f3c57fe6951a","e-1296c7983fa47a9a"],"text":"A063007 \u2192 basis A013610 \u2192 D"},"headline":"A063007 \u2192 basis A013610 \u2192 D","chain":null}