{"id":"n-3e24a2a5ce552865","revision":0,"first_index":0,"fingerprint":"f6-723ac5bf98d71a32","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","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":true,"rows":[["529","1104","576"],["12167","38088","39744","13824"],["279841","1168032","1828224","1271808","331776"],["6436343","33580920","70081920","73128960","38154240","7962624"],["148035889","926833392","2417826240","3363932160","2632642560","1098842112","191102976"],["3404825447","24870029352","77854004928","135398269440","141285150720","88456790016","30767579136","4586471424"],["78310985281","653726485824","2387522817792","4982656315392","6499116933120","5425349787648","2830617280512","843910742016","110075314176"],["1801152661463","16915172820696","70602460468992","171901642881024","269063441031168","280761851510784","195312592355328","87344761798656","22785590034432","2641807540224"],["41426511213649","432276638751120","2029820738483520","5648196837519360","10314098572861440","12915045169496064","11230474060431360","6696431737896960","2620342853959680","607615734251520","63403380965376"],["952809757913927","10936598960403336","57060516315147840","178624224986549760","372780991276277760","544584404647084032","568261987457826816","423549307421982720","220982247350599680","76863390382817280","16041055384240128","1521681143169024"],["21914624432020321","274409210279210976","1574870250298080384","5477809566254192640","12860944199031582720","21472185097513598976","26140051423060033536","23379921769693446144","15247775067191377920","7071431915219189760","2213665643025137664","419983995514650624","36520347436056576"],["504036361936467383","6837362822790340152","42807836803556912256","163786506031000359936","427269146167827025920","802522918019570761728","1116553625070707146752","1165099434856390066176","911816949018044399616","528589535662634434560","220628675754838720512","62787607329440268288","10919583883380916224","876488338465357824"],["11592836324538749809","169356217610653040688","1148676954228777145536","4794477721998374172672","13758066506604030234624","28712486622477976141824","44941283409095962656768","53594574003393943044096","48934176263968382779392","34041166096673657585664","17760608398264517001216","6739203186693255462912","1758053005224327512064","282229244985845219328","21035720123168587776"],["266635235464391245607","4173421076833949931240","30484119169917547323840","137841234507453257464320","431502994979853675540480","990580788475490176892928","1722749197348678568509440","2311266003896363793776640","2411755830152727436984320","1957367050558735311175680","1225481979480251673083904","581256274852293283676160","202176095600797663887360","48684544760058300334080","7257323442493162782720","504857282956046106624"],["6132610415680998648961","102387930418326238313088","801296846752118386798080","3901967253749446057451520","13232758512715512716574720","33139430014452762281508864","63397170462431371321147392","94505098825984652901089280","110940768187025462101278720","102901582086516370644664320","75162894741455435949146112","42780461829128785678565376","18600200795273385077637120","5971970823900484840980480","1335347513418741952020480","185787480127824967237632","12116574790945106558976"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2209","103823","4879681","229345007","10779215329","506623120463","23811286661761","1119130473102767","52599132235830049","2472159215084012303","116191483108948578241","5460999706120583177327","256666986187667409334369","12063348350820368238715343","566977372488557307219621121"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["529","1104","576"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["23","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201102,"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-5a234ac48b943e8b"]},{"id":201103,"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-5a234ac48b943e8b"]},{"id":201104,"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-5a234ac48b943e8b"]}],"region_exact":{},"incoming_edges":[{"id":"e-5a234ac48b943e8b","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3e24a2a5ce552865","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":"cf835dbca3bdb32299d914f3c0951c1290c16f8d1e41e37fea4205350168c791","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f55a7ba0b32633bc","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-708acfa3803762f7","kind":"fingerprint_related","node_a_id":"n-3e24a2a5ce552865","node_b_id":"n-f7d94fed0ec64a12","fingerprint":"f6-723ac5bf98d71a32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-723ac5bf98d71a32"},"other_id":"n-f7d94fed0ec64a12","other":{"id":"n-f7d94fed0ec64a12","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-723ac5bf98d71a32","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2116","4416","2304"],["97336","304704","317952","110592"],["4477456","18688512","29251584","20348928","5308416"]],"card_rows":[["2116","4416","2304"],["97336","304704","317952","110592"],["4477456","18688512","29251584","20348928","5308416"],["205962976","1074589440","2242621440","2340126720","1220935680","254803968"]],"row_sums_preview":["8836","830584","78074896","7339040224","689869781056","64847759419264","6095689385410816","572994802228616704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013609","basis A013609"],"edges":["e-61350e73fec19600","e-4d5cc40657cd7de6"],"text":"A038314 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038314 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-a5c23d57215c389c","kind":"fingerprint_related","node_a_id":"n-3afaca41e5482e8c","node_b_id":"n-3e24a2a5ce552865","fingerprint":"f6-723ac5bf98d71a32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-723ac5bf98d71a32"},"other_id":"n-3afaca41e5482e8c","other":{"id":"n-3afaca41e5482e8c","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-723ac5bf98d71a32","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["23","24"],["529","1104","576"]],"card_rows":[["1"],["23","24"],["529","1104","576"],["12167","38088","39744","13824"]],"row_sums_preview":["1","47","2209","103823","4879681","229345007","10779215329","506623120463"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["basis A013609"],"edges":["e-15af34f262d68ea6"],"text":"A038326 \u2192 basis A013609"},"headline":"A038326 \u2192 basis A013609","chain":null}},{"id":"l-ad2e4b72667385f0","kind":"fingerprint_related","node_a_id":"n-55fdb7333755864e","node_b_id":"n-3e24a2a5ce552865","fingerprint":"f6-723ac5bf98d71a32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-723ac5bf98d71a32"},"other_id":"n-55fdb7333755864e","other":{"id":"n-55fdb7333755864e","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-723ac5bf98d71a32","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["24"],["1104","1152"]],"card_rows":[["24"],["1104","1152"],["38088","79488","41472"],["1168032","3656448","3815424","1327104"]],"row_sums_preview":["0","24","2256","159048","9967008","585561720","33025681008","1810908175272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["basis A013609","D"],"edges":["e-15af34f262d68ea6","e-c652f7bfa7ae035f"],"text":"A038326 \u2192 basis A013609 \u2192 D"},"headline":"A038326 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-db1576eb1c434a0d","kind":"fingerprint_related","node_a_id":"n-1473f71bb1b53b4e","node_b_id":"n-3e24a2a5ce552865","fingerprint":"f6-723ac5bf98d71a32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-723ac5bf98d71a32"},"other_id":"n-1473f71bb1b53b4e","other":{"id":"n-1473f71bb1b53b4e","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-723ac5bf98d71a32","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","23"],["576","1104","529"]],"card_rows":[["1"],["24","23"],["576","1104","529"],["13824","39744","38088","12167"]],"row_sums_preview":["1","47","2209","103823","4879681","229345007","10779215329","506623120463"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["basis A013609","reverse"],"edges":["e-15af34f262d68ea6","e-1251e689027c617a"],"text":"A038326 \u2192 basis A013609 \u2192 reverse"},"headline":"A038326 \u2192 basis A013609 \u2192 reverse","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:             529,              1104,               576;\n  1:           12167,             38088,             39744,              13824;\n  2:          279841,           1168032,           1828224,            1271808,             331776;\n  3:         6436343,          33580920,          70081920,           73128960,           38154240,            7962624;\n  4:       148035889,         926833392,        2417826240,         3363932160,         2632642560,         1098842112,          191102976;\n  5:      3404825447,       24870029352,       77854004928,       135398269440,       141285150720,        88456790016,        30767579136,         4586471424;\n  6:     78310985281,      653726485824,     2387522817792,      4982656315392,      6499116933120,      5425349787648,      2830617280512,       843910742016,       110075314176;\n  7:   1801152661463,    16915172820696,    70602460468992,    171901642881024,    269063441031168,    280761851510784,    195312592355328,     87344761798656,     22785590034432,     2641807540224;\n  8:  41426511213649,   432276638751120,  2029820738483520,   5648196837519360,  10314098572861440,  12915045169496064,  11230474060431360,   6696431737896960,   2620342853959680,   607615734251520,    63403380965376;\n  9: 952809757913927, 10936598960403336, 57060516315147840, 178624224986549760, 372780991276277760, 544584404647084032, 568261987457826816, 423549307421982720, 220982247350599680, 76863390382817280, 16041055384240128, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1104*x + 576*x^2.","mathematica":"P[0, x_] := 529 + 1104*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[(23 + 24*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] = 529 + 1104*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 24*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-108b34f10b7468f4","steps":["basis A013611","basis A013609"],"edges":["e-8f18b93eb5dfe7b3","e-5a234ac48b943e8b"],"text":"A038281 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038281 \u2192 basis A013611 \u2192 basis A013609","chain":null}