{"id":"n-302c04144b6d1b7d","revision":0,"first_index":0,"fingerprint":"f6-dceb48071bfd4833","name":"basis_map_A013610 of basis_map_A013610 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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"],["5","9"],["29","99","81"],["185","918","1458","729"],["1257","8154","19116","18954","6561"],["8925","71415","221130","328050","229635","59049"],["65445","622485","2394765","4734855","5019165","2657205","531441"],["491825","5419260","24913980","61593210","87720570","71094996","29760696","4782969"],["3768209","47193156","252355824","749079576","1341619524","1471501080","952342272","325241892","43046721"],["29324405","411368949","2508200316","8691470424","18770732316","26025728652","22945496616","12225268764","3486784401","387420489"],["231153133","3590171415","24582170385","97410838950","246410296020","413589352428","463150831500","338203737990","151739691525","36804946455","3486784401"],["1841801065","31373848242","238345003710","1063145107035","3083601962430","6080854210200","8273235402432","7707228416910","4765917715515","1832498912970","383546284110","31381059609"],["14810069497","274528881342","2291434799148","11362602801330","37178465923665","84272221327932","135090788656032","153412831476828","121520247682185","64734089507010","21632010423804","3954013510734","282429536481"],["120029657805","2405225544579","21879498090294","119400494661162","435105799700715","1114855581888477","2058949479473172","2762251092711396","2675597569824465","1832724779115087","852623389576530","250483617799038","40387423716783","2541865828329"],["979470140661","21098056046697","207742433802813","1237283128664037","4969660096253853","14205794960758401","29713208445869949","46024408663000911","52796556605741949","44353202800347441","26627278873424229","10940096381949189","2852820747994581","409240398360969","22876792454961"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","209","3290","54042","918204","16025361","285777506","5186148254","95481795332","1779199974202","33492969724228","636020474595988","12169325847587832","234382247854810641"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","9"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["9","54","81"],["-9","-54","-81"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","9"],["10","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202879,"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-ca4423255a30137c"]},{"id":202880,"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-ca4423255a30137c"]},{"id":202881,"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-ca4423255a30137c"]},{"id":202882,"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-ca4423255a30137c"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca4423255a30137c","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-302c04144b6d1b7d","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":"cc4a44cd67c6d8542a0b7eae648e96892d9802ed0e445af558bc9a1f9631a4ed","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-169aeabe58b22027","role":"sequence","name":"basis_map_A013610 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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-370a6834bd6a02f9","kind":"fingerprint_related","node_a_id":"n-302c04144b6d1b7d","node_b_id":"n-8a459424c405b599","fingerprint":"f6-dceb48071bfd4833","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dceb48071bfd4833"},"other_id":"n-8a459424c405b599","other":{"id":"n-8a459424c405b599","name":"basis_map_A013610 of basis_map_A013610 of narayana","first_index":1,"fingerprint":"f6-dceb48071bfd4833","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","9"],["29","99","81"]],"card_rows":[["1"],["5","9"],["29","99","81"],["185","918","1458","729"]],"row_sums_preview":["1","14","209","3290","54042","918204","16025361","285777506"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013610","basis A013610"],"edges":["e-21b441a6a6a148f7","e-a4054c866ebc4d54"],"text":"narayana \u2192 basis A013610 \u2192 basis A013610"},"headline":"narayana \u2192 basis A013610 \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:        5,         9;\n  2:       29,        99,         81;\n  3:      185,       918,       1458,        729;\n  4:     1257,      8154,      19116,      18954,        6561;\n  5:     8925,     71415,     221130,     328050,      229635,       59049;\n  6:    65445,    622485,    2394765,    4734855,     5019165,     2657205,      531441;\n  7:   491825,   5419260,   24913980,   61593210,    87720570,    71094996,    29760696,     4782969;\n  8:  3768209,  47193156,  252355824,  749079576,  1341619524,  1471501080,   952342272,   325241892,   43046721;\n  9: 29324405, 411368949, 2508200316, 8691470424, 18770732316, 26025728652, 22945496616, 12225268764, 3486784401, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(9*n-9+(54*n-54)*x+(81*n-81)*x^2)*P(n-2,x) + (10*n+5+(18*n+9)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 9*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 9*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(9*n - 9 + (54*n - 54)*x + (81*n - 81)*x^2)*P[n - 2, x] + (10*n + 5 + (18*n + 9)*x)*P[n - 1, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n - 9 + (54*n - 54)*x + (81*n - 81)*x^2)*v[i - 2] + (10*n + 5 + (18*n + 9)*x)*v[i - 1])/(n + 2)); 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-c2eaf5212a4dec13","steps":["basis A013610","basis A013610"],"edges":["e-9081e2b3ba0d880f","e-ca4423255a30137c"],"text":"A001263 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A001263 \u2192 basis A013610 \u2192 basis A013610","chain":null}