{"id":"n-13114c927e22057d","revision":1,"first_index":0,"fingerprint":"f6-97fb096e972a926e","name":"Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["0","1"],["-1","1","1"],["-1","-2","3","1"],["2","-9","1","6","1"],["9","-9","-25","15","10","1"],["9","50","-104","-20","50","15","1"],["-50","267","-98","-364","105","119","21","1"],["-267","413","1163","-1610","-539","574","238","28","1"],["-413","-2180","7569","-1511","-6636","903","1806","426","36","1"],["2180","-17731","17491","29580","-32570","-12957","8757","4500","705","45","1"],["17731","-50533","-71489","236511","-35915","-149897","8085","34617","9735","1100","55","1"],["50533","110176","-903044","781704","842336","-833910","-343706","172524","102597","19085","1639","66","1"],["-110176","1966797","-4041024","-1927276","8320598","-1264978","-4103814","43186","821535","257972","34749","2353","78","1"],["-1966797","9938669","-333423","-43105244","37680006","26618592","-26190164","-10373792","4293718","2795793","579579","59696","3276","91","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","1","1","1","1","1","1","1","1","1","1","1","1","1"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A341287","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160450,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]","region_exact":null,"edges":["e-0f172f031b36afc5","e-73143461db750abd"]},{"id":160451,"position":1,"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-73143461db750abd"]}],"region_exact":{},"incoming_edges":[{"id":"e-0f172f031b36afc5","operation":{"map":"A341287","operation":"basis_map"},"operation_detail":"A341287","conclusion_id":"n-13114c927e22057d","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; 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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":"d6c1c1661e74d970b0e4b8b5ee50d02f5aaefaadeb4f91521147f4398530a7c9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3cb22164e2f83fe1","role":"sequence","name":"Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= 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":[{"id":"e-cb47efc3e6c3b2c6","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-24c1a22c5b1801ba","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise)."},{"id":"e-cb47efc3e6c3b2c6","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-24c1a22c5b1801ba","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise)."}],"used_by_total":2,"links":[{"id":"l-5edf22c5fd82ccfe","kind":"oeis_match","node_a_id":"n-13114c927e22057d","node_b_id":null,"fingerprint":null,"oeis_id":"A341287","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A341287","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A341287","relation":"identification"},{"oeis_id":"A341287","relation":"match_triangle"}],"bfiles":[{"node_id":"n-13114c927e22057d","oeis_id":"A341287","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":22217}],"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:    0,     1;\n  2:   -1,     1,    1;\n  3:   -1,    -2,    3,     1;\n  4:    2,    -9,    1,     6,     1;\n  5:    9,    -9,  -25,    15,    10,   1;\n  6:    9,    50, -104,   -20,    50,  15,    1;\n  7:  -50,   267,  -98,  -364,   105, 119,   21,   1;\n  8: -267,   413, 1163, -1610,  -539, 574,  238,  28,  1;\n  9: -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (1 - x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] - (1 - x)*deriv(v[i - 1], x)); 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}},{"oeis_id":"A341287","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    0,     1;\n  2:   -1,     1,    1;\n  3:   -1,    -2,    3,     1;\n  4:    2,    -9,    1,     6,     1;\n  5:    9,    -9,  -25,    15,    10,   1;\n  6:    9,    50, -104,   -20,    50,  15,    1;\n  7:  -50,   267,  -98,  -364,   105, 119,   21,   1;\n  8: -267,   413, 1163, -1610,  -539, 574,  238,  28,  1;\n  9: -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (1 - x)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] - (1 - x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["basis A341287"],"edges":["e-0f172f031b36afc5"],"text":"A023531 \u2192 basis A341287"},"headline":"A341287: Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).","chain":"A023531 \u2192 basis A341287"}