{"id":"n-a6c3537128a476c6","revision":1,"first_index":1,"fingerprint":"f6-ff058243d1bd7929","name":"Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["1"],["1","-1"],["1","-3","2"],["1","-6","11","-6"],["1","-10","35","-50","24"],["1","-15","85","-225","274","-120"],["1","-21","175","-735","1624","-1764","720"],["1","-28","322","-1960","6769","-13132","13068","-5040"],["1","-36","546","-4536","22449","-67284","118124","-109584","40320"],["1","-45","870","-9450","63273","-269325","723680","-1172700","1026576","-362880"],["1","-55","1320","-18150","157773","-902055","3416930","-8409500","12753576","-10628640","3628800"],["1","-66","1925","-32670","357423","-2637558","13339535","-45995730","105258076","-150917976","120543840","-39916800"],["1","-78","2717","-55770","749463","-6926634","44990231","-206070150","657206836","-1414014888","1931559552","-1486442880","479001600"],["1","-91","3731","-91091","1474473","-16669653","135036473","-790943153","3336118786","-9957703756","20313753096","-26596717056","19802759040","-6227020800"],["1","-105","5005","-143325","2749747","-37312275","368411615","-2681453775","14409322928","-56663366760","159721605680","-310989260400","392156797824","-283465647360","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A008276","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160478,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, +inf)","region_exact":null,"edges":["e-68dd2f09e1ded893"]},{"id":160479,"position":1,"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-68dd2f09e1ded893"]}],"region_exact":{},"incoming_edges":[{"id":"e-68dd2f09e1ded893","operation":{"index_shift":-1,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift -1","conclusion_id":"n-a6c3537128a476c6","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":"b699524ecaf15f5e374ddb3cc307097038d07e54b46bfc2858c226e16ab68d7f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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":[{"id":"e-ed573324c4b51ce2","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-ed9cc69a9e3761ed","conclusion_name":"obreschkoff_combination of Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x."},{"id":"e-ed573324c4b51ce2","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-ed9cc69a9e3761ed","conclusion_name":"obreschkoff_combination of Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x."}],"used_by_total":2,"links":[{"id":"l-f3d09066b0555afc","kind":"fingerprint_related","node_a_id":"n-a52b649d7348b7a5","node_b_id":"n-a6c3537128a476c6","fingerprint":"f6-ff058243d1bd7929","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ff058243d1bd7929"},"other_id":"n-a52b649d7348b7a5","other":{"id":"n-a52b649d7348b7a5","name":"Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.","first_index":1,"fingerprint":"f6-ff058243d1bd7929","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","-1","1"]],"card_rows":[["1"],["0","1"],["0","-1","1"],["0","2","-3","1"]],"row_sums_preview":["1","1","0","0","0","0","0","0"],"regions":["[-1/2, +inf)"],"region_exact":{},"oeis_ids":["A048994"],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 \u22122x\u22121 (shift -1)"],"edges":["e-6e459812af5dc78f"],"text":"A028338 \u2192 x \u21a6 \u22122x\u22121 (shift -1)"},"headline":"A048994: Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.","chain":"A028338 \u2192 x \u21a6 \u22122x\u22121 (shift -1)"}},{"id":"l-2a5f21512c103228","kind":"oeis_match","node_a_id":"n-a6c3537128a476c6","node_b_id":null,"fingerprint":null,"oeis_id":"A008276","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":47},"in_catalog":true,"oeis_id":"A008276","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-f37a57db13698892","kind":"oeis_match","node_a_id":"n-a6c3537128a476c6","node_b_id":null,"fingerprint":null,"oeis_id":"A008275","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":false,"oeis_id":"A008275","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008276","relation":"identification"},{"oeis_id":"A008275","relation":"match_reversed_triangle"},{"oeis_id":"A008276","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a6c3537128a476c6","oeis_id":"A008275","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":27358},{"node_id":"n-a6c3537128a476c6","oeis_id":"A008276","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":47,"oeis_terms_agreeing":47,"oeis_terms_available":47,"term_count":990,"raw_size":27358}],"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: 1,  -1;\n   3: 1,  -3,   2;\n   4: 1,  -6,  11,    -6;\n   5: 1, -10,  35,   -50,    24;\n   6: 1, -15,  85,  -225,   274,    -120;\n   7: 1, -21, 175,  -735,  1624,   -1764,    720;\n   8: 1, -28, 322, -1960,  6769,  -13132,  13068,    -5040;\n   9: 1, -36, 546, -4536, 22449,  -67284, 118124,  -109584,   40320;\n  10: 1, -45, 870, -9450, 63273, -269325, 723680, -1172700, 1026576, -362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];\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] = (1 - (n - 1)*x)*v[i - 1]); 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}},{"oeis_id":"A008275","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:      -1,       1;\n   3:       2,      -3,        1;\n   4:      -6,      11,       -6,      1;\n   5:      24,     -50,       35,    -10,       1;\n   6:    -120,     274,     -225,     85,     -15,     1;\n   7:     720,   -1764,     1624,   -735,     175,   -21,     1;\n   8:   -5040,   13068,   -13132,   6769,   -1960,   322,   -28,   1;\n   9:   40320, -109584,   118124, -67284,   22449, -4536,   546, -36,   1;\n  10: -362880, 1026576, -1172700, 723680, -269325, 63273, -9450, 870, -45, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 1)*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A008276","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  -1;\n   3: 1,  -3,   2;\n   4: 1,  -6,  11,    -6;\n   5: 1, -10,  35,   -50,    24;\n   6: 1, -15,  85,  -225,   274,    -120;\n   7: 1, -21, 175,  -735,  1624,   -1764,    720;\n   8: 1, -28, 322, -1960,  6769,  -13132,  13068,    -5040;\n   9: 1, -36, 546, -4536, 22449,  -67284, 118124,  -109584,   40320;\n  10: 1, -45, 870, -9450, 63273, -269325, 723680, -1172700, 1026576, -362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 1)*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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":true}}],"derivation":{"root":"n-7fd16a32c438d522","steps":["x \u21a6 \u2212x\u22121 (shift -1)"],"edges":["e-68dd2f09e1ded893"],"text":"A059364 \u2192 x \u21a6 \u2212x\u22121 (shift -1)"},"headline":"A008276: Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.","chain":"A059364 \u2192 x \u21a6 \u2212x\u22121 (shift -1)"}