{"id":"n-bc4bccd8525de23a","revision":0,"first_index":0,"fingerprint":"f6-23394a58ed9a04a5","name":"basis_map_A013609 of derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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":false,"rows":[[],["1"],["7","8"],["46","116","72"],["326","1316","1752","768"],["2556","14416","30252","27968","9600"],["22212","162256","471252","679808","486720","138240"],["212976","1919984","7177392","14235168","15790400","9282816","2257920"],["2239344","24095088","110676144","281234208","426816960","386721024","193600512","41287680"],["25659360","321567264","1757328816","5468569536","10595939280","13086456576","10057146624","4395515904","836075520"],["318540960","4563614304","28977634896","107016366656","253269570480","398253565056","415979266304","278228312064","108095938560","18579456000"],["4261576320","68779173888","498330890784","2134207023488","5982184928800","11465359441344","15213592698752","13797647011840","8183301986304","2865298636800","449622835200"],["61148511360","1098684923904","8954351840160","43691279810432","141794614779040","321336598686912","518806769441408","596656525176832","478919395141632","255470268579840","81488980869120","11771943321600"],["937030429440","18561311525376","168209637794496","922095447804928","3405091200745600","8922704362722816","17010504932349376","23769163930163200","24157074893405184","17412032953098240","8447295413551104","2476131027517440","331576403558400"],["15275952518400","330893376929280","3302719650983232","20111258194735104","83349708684356480","248251629785894400","546632583167519296","900873616787628032","1111084787201218560","1012733935424634880","663000046158692352","295168443499413504","80082953888071680","9998303861145600"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","15","234","4162","84792","1960488","50876656","1466670960","46544258880","1613282265280","60662586203520","2460050561082240","107041378544665600","4974936155633740800"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-4"],["0","4","8"],["-1","-4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","-4"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200001,"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-5892d22641c77abc"]},{"id":200002,"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-5892d22641c77abc"]},{"id":200003,"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-5892d22641c77abc"]},{"id":200004,"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-5892d22641c77abc"]}],"region_exact":{},"incoming_edges":[{"id":"e-5892d22641c77abc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-bc4bccd8525de23a","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":"fcb6e030b25505d6ddc80d4a6e1153423dc3ecfb3634de8bbde280c257bbb2ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-89353a17ee1d740a","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-32964b3d5593a7fe","kind":"fingerprint_related","node_a_id":"n-a63ba5af2d4957e9","node_b_id":"n-bc4bccd8525de23a","fingerprint":"f6-23394a58ed9a04a5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-23394a58ed9a04a5"},"other_id":"n-a63ba5af2d4957e9","other":{"id":"n-a63ba5af2d4957e9","name":"derivative of basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-23394a58ed9a04a5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["14","16"]],"card_rows":[["2"],["14","16"],["92","232","144"],["652","2632","3504","1536"]],"row_sums_preview":["0","2","30","468","8324","169584","3920976","101753312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["basis A013609","D"],"edges":["e-23134a27b4e4bcc8","e-ad47fe23ef235a42"],"text":"A094638 \u2192 basis A013609 \u2192 D"},"headline":"A094638 \u2192 basis A013609 \u2192 D","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:        1;\n  2:        7,         8;\n  3:       46,       116,         72;\n  4:      326,      1316,       1752,        768;\n  5:     2556,     14416,      30252,      27968,        9600;\n  6:    22212,    162256,     471252,     679808,      486720,      138240;\n  7:   212976,   1919984,    7177392,   14235168,    15790400,     9282816,     2257920;\n  8:  2239344,  24095088,  110676144,  281234208,   426816960,   386721024,   193600512,   41287680;\n  9: 25659360, 321567264, 1757328816, 5468569536, 10595939280, 13086456576, 10057146624, 4395515904, 836075520;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (4*n^2 - 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (4*n^2 - 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-4d60c3dd8627c01d","steps":["D","basis A013609"],"edges":["e-74e2b15799b20e69","e-5892d22641c77abc"],"text":"A094638 \u2192 D \u2192 basis A013609"},"headline":"A094638 \u2192 D \u2192 basis A013609","chain":null}