{"id":"n-3ce0bead7a116750","revision":0,"first_index":1,"fingerprint":"f6-e4c3b23b17de5bef","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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"],["2","1"],["6","7","2"],["22","41","25","5"],["90","231","219","91","14"],["394","1289","1666","1064","336","42"],["1806","7183","11780","10200","4920","1254","132"],["8558","40081","79731","87345","56925","22077","4719","429"],["41586","224143","524405","695695","572495","299299","97097","17875","1430"],["206098","1256465","3380740","5269880","5245240","3457454","1509508","420992","68068","4862"],["1037718","7059735","21477510","38480520","44968560","35818146","19695312","7383168","1806012","260338","16796"],["5293446","39753273","134930445","273241605","366832830","342842136","227631768","107379480","35271600","7684170","999362","58786"],["27297738","224298231","840313551","1898128155","2879299500","3090172008","2406182856","1369724136","565779720","165392150","32481526","3848222","208012"],["142078746","1267854873","5196728070","12954973680","21915622080","26568329436","23747740368","15846948480","7895121300","2900633670","763909212","136574736","14858000","742900"],["745387038","7178461215","31954358280","87146576400","162678824880","219893080836","221958837672","169969607808","99230045772","43953544778","14541405736","3484498160","571735840","57500460","2674440"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","93","645","4791","37275","299865","2474025","20819307","178003815","1541918901","13503125805","119352115551","1063366539315"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-907ff3ac724c18dd","recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["3"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-9","-6"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175650,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-9c91cf7fc3768a40"]},{"id":175651,"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-8d157ce2d69081c8","e-9c91cf7fc3768a40"]},{"id":175652,"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-8d157ce2d69081c8","e-9c91cf7fc3768a40"]},{"id":175653,"position":3,"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-8d157ce2d69081c8"]},{"id":175654,"position":4,"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-8d157ce2d69081c8"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d157ce2d69081c8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3ce0bead7a116750","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":"10a002443a6509662a13991e326a392998bc3180135d5f01206f274e6c9dcf3d","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-be392fc9b98fb771","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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]"}]},{"id":"e-907ff3ac724c18dd","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-3ce0bead7a116750","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"f67130aa94047df16c0fff2c11f26681d9cd1d57b19bdcadc347707bd85c62bb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-9c91cf7fc3768a40","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3ce0bead7a116750","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"605c06d5679c63cfe445bb6382b0c0b831e86600b73f147c75eb9b471d8b088a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-be392fc9b98fb771","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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-cb28289405a468c1","kind":"fingerprint_related","node_a_id":"n-3ce0bead7a116750","node_b_id":"n-07d0d70f782a78c7","fingerprint":"f6-e4c3b23b17de5bef","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e4c3b23b17de5bef"},"other_id":"n-07d0d70f782a78c7","other":{"id":"n-07d0d70f782a78c7","name":"shift_x_plus_one of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-e4c3b23b17de5bef","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["36","42","12"]],"card_rows":[["1"],["4","2"],["36","42","12"],["528","984","600","120"]],"row_sums_preview":["1","6","90","2232","77400","3449520","187866000","12090556800"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-694017a14c5c1f2e","steps":["basis A007318"],"edges":["e-7c0eb5117c2561c6"],"text":"A253284 \u2192 basis A007318"},"headline":"A253284 \u2192 basis A007318","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:      1;\n   2:      2,       1;\n   3:      6,       7,       2;\n   4:     22,      41,      25,       5;\n   5:     90,     231,     219,      91,      14;\n   6:    394,    1289,    1666,    1064,     336,      42;\n   7:   1806,    7183,   11780,   10200,    4920,    1254,     132;\n   8:   8558,   40081,   79731,   87345,   56925,   22077,    4719,    429;\n   9:  41586,  224143,  524405,  695695,  572495,  299299,   97097,  17875,  1430;\n  10: 206098, 1256465, 3380740, 5269880, 5245240, 3457454, 1509508, 420992, 68068, 4862;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n-2+(2*n-2)*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 2 + (2*n - 2)*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 2 + (2*n - 2)*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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-be392fc9b98fb771","steps":["basis A007318"],"edges":["e-8d157ce2d69081c8"],"text":"A088617 \u2192 basis A007318"},"headline":"A088617 \u2192 basis A007318","chain":null}