{"id":"n-ee8b5acfa7aa582c","revision":0,"first_index":0,"fingerprint":"f6-c93eb9c90ec44c16","name":"basis_map_A013610 of basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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"],["6","12"],["48","168","144"],["480","2256","3456","1728"],["5760","32832","68544","62208","20736"],["80640","528768","1353600","1693440","1036800","248832"],["1290240","9427968","28002816","43338240","36910080","16422912","2985984"],["23224320","185186304","617186304","1116122112","1184440320","738533376","250822656","35831808"],["464486400","3982417920","14565961728","29728677888","37082271744","28983951360","13878853632","3726508032","429981696"],["10218700800","93187031040","368240173056","828822454272","1172554113024","1082634190848","653142196224","248529420288","54177693696","5159780352"],["245248819200","2359113154560","9956008525824","24310620979200","38087168163840","40053869936640","28667022999552","13802412441600","4282617692160","773967052800","61917364224"],["6376469299200","64279927848960","287165579526144","751548247769088","1281993824010240","1498446636318720","1225989037228032","702866999476224","276977009295360","71514555678720","10897456103424","743008370688"],["178541140377600","1876355611361280","8811995360919552","24489337891848192","44914406045515776","57340431705047040","52309052678209536","34392144432070656","16189760253984768","5326131670548480","1163303439040512","151573707620352","8916100448256"],["5356234211328000","58433162025369600","286876128163921920","840424081086480384","1641304236067651584","2259185823697600512","2257356760806850560","1659472965100634112","898398540804390912","354061073164271616","98812683217797120","18506852497096704","2086367504891904","106993205379072"],["171399494762496000","1934135995347763200","9881234045549936640","30336084132734435328","62606824527202615296","91989597191135035392","99345646230190424064","80191416012902498304","48662428886948118528","22110736830909382656","7410738740940767232","1777971478520659968","288845990121701376","28460192630833152","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","360","7920","190080","4942080","138378240","4151347200","132843110400","4516665753600","162599967129600","6178798750924800","247151950036992000","10380381901553664000","456736803668361216000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","12"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203416,"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-a67c823f525cc289"]},{"id":203417,"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-a67c823f525cc289"]},{"id":203418,"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-a67c823f525cc289"]},{"id":203419,"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-a67c823f525cc289"]}],"region_exact":{},"incoming_edges":[{"id":"e-a67c823f525cc289","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-ee8b5acfa7aa582c","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"574f88c0973711a4fb58f2b88a352d7f5c99bf6f3a827cbea67c058357b91d13","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-dcd6be200625a597","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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-f27714bf22f15e91","kind":"fingerprint_related","node_a_id":"n-2d0bef3d53a72546","node_b_id":"n-ee8b5acfa7aa582c","fingerprint":"f6-c93eb9c90ec44c16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c93eb9c90ec44c16"},"other_id":"n-2d0bef3d53a72546","other":{"id":"n-2d0bef3d53a72546","name":"basis_map_A013610 of basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-c93eb9c90ec44c16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["12","42","36"]],"card_rows":[["1"],["3","6"],["12","42","36"],["60","282","432","216"]],"row_sums_preview":["1","9","90","990","11880","154440","2162160","32432400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A013609","basis A013610"],"edges":["e-5d127e92ec7ea44a","e-ed5dcaa220cbb867"],"text":"A132393 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A132393 \u2192 basis A013609 \u2192 basis A013610","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:           1;\n  1:           6,          12;\n  2:          48,         168,          144;\n  3:         480,        2256,         3456,         1728;\n  4:        5760,       32832,        68544,        62208,         20736;\n  5:       80640,      528768,      1353600,      1693440,       1036800,        248832;\n  6:     1290240,     9427968,     28002816,     43338240,      36910080,      16422912,      2985984;\n  7:    23224320,   185186304,    617186304,   1116122112,    1184440320,     738533376,    250822656,     35831808;\n  8:   464486400,  3982417920,  14565961728,  29728677888,   37082271744,   28983951360,  13878853632,   3726508032,   429981696;\n  9: 10218700800, 93187031040, 368240173056, 828822454272, 1172554113024, 1082634190848, 653142196224, 248529420288, 54177693696, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+4+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 4 + 12*x)*P[n - 1, 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] = (2*n + 4 + 12*x)*v[i - 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-4f6e747b794bcc2e","steps":["basis A013611","basis A013610"],"edges":["e-38e97daa1f393e3a","e-a67c823f525cc289"],"text":"A028338 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A028338 \u2192 basis A013611 \u2192 basis A013610","chain":null}
