{"id":"n-401d70b5bb38eaa4","revision":0,"first_index":0,"fingerprint":"f6-fb89365bfc869b8b","name":"basis_map_A013611 of one_plus_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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["13","44","32"],["70","416","752","384"],["446","3936","12272","15488","6144"],["3276","39056","178672","385088","377344","122880"],["27252","413360","2534032","7951808","13225984","10653696","2949120"],["253296","4691872","36389504","152160768","366527744","499167232","342540288","82575360"],["2602224","57147552","539084928","2840146432","9078945024","17838329856","20640481280","12365070336","2642411520"],["29288160","745771968","8315584320","53120976640","213283584768","554394633216","922835124224","930430255104","495348350976","95126814720"],["358457760","10401392448","134164996800","1010678319360","4908444060928","15985964583936","35118749097984","50826179969024","45479172243456","21811597148160","3805072588800"],["4740577920","154604653056","2268440258688","19731420938752","112788615048704","443463469211648","1218264267468800","2322481636212736","2977974717251584","2398110816927744","1047226944061440","167423193907200"],["67375532160","2441963893248","40212147056256","397286781630976","2618244074948096","12097742227152896","40058814016237568","95345539933634560","160789382767575040","185291162026770432","135765027415130112","54452473162629120","8036313307545600"],["1024208720640","40870744768512","747083797279488","8273881650676736","61858222794369024","329111706976673792","1279023342075113472","3664551449158238208","7705437664242171904","11663499170969288704","12216014852270850048","8216581981983473664","3048509063760445440","417888291992371200"],["16583626886400","722890563932160","14534452111650048","178504570363124736","1494485404507584512","9010928424308383744","40280520091486982144","135302957418942808064","342093616997920145408","644936532947862814720","886319957241466191872","851382080383708299264","529521436554474553344","182835268038363709440","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","89","1622","38286","1106316","37755252","1484306064","66004219152","3273630384096","179090782858656","10709892866518272","694894707199736064","48611538606624440832","3646773306351279853056"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-16"],["0","8","32"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","-8"],["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205622,"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-98a7aea6afd6fa69"]},{"id":205623,"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-98a7aea6afd6fa69"]},{"id":205624,"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-98a7aea6afd6fa69"]},{"id":205625,"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-98a7aea6afd6fa69"]}],"region_exact":{},"incoming_edges":[{"id":"e-98a7aea6afd6fa69","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-401d70b5bb38eaa4","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":"43ac9e452fc4dd9eecf3e5b7e63454ce4ade4502c8f1276d71384143520c567c","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-51d4e271f3dd1e2d","role":"sequence","name":"one_plus_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":[],"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:        3,         4;\n  2:       13,        44,         32;\n  3:       70,       416,        752,         384;\n  4:      446,      3936,      12272,       15488,         6144;\n  5:     3276,     39056,     178672,      385088,       377344,       122880;\n  6:    27252,    413360,    2534032,     7951808,     13225984,     10653696,      2949120;\n  7:   253296,   4691872,   36389504,   152160768,    366527744,    499167232,    342540288,     82575360;\n  8:  2602224,  57147552,  539084928,  2840146432,   9078945024,  17838329856,  20640481280,  12365070336,   2642411520;\n  9: 29288160, 745771968, 8315584320, 53120976640, 213283584768, 554394633216, 922835124224, 930430255104, 495348350976, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*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] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*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+1","basis A013611"],"edges":["e-55797f5beaf34466","e-98a7aea6afd6fa69"],"text":"A094638 \u2192 D+1 \u2192 basis A013611"},"headline":"A094638 \u2192 D+1 \u2192 basis A013611","chain":null}
