{"id":"n-f45d29a8d57e6cc8","revision":1,"first_index":0,"fingerprint":"f6-0f5f9e74c0cf88d5","name":"Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","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":true,"rows":[["1"],["-10","1"],["200","-30","1"],["-6000","1100","-60","1"],["240000","-50000","3500","-100","1"],["-12000000","2740000","-225000","8500","-150","1"],["720000000","-176400000","16240000","-735000","17500","-210","1"],["-50400000000","13068000000","-1313200000","67690000","-1960000","32200","-280","1"],["4032000000000","-1095840000000","118124000000","-6728400000","224490000","-4536000","54600","-360","1"],["-362880000000000","102657600000000","-11727000000000","723680000000","-26932500000","632730000","-9450000","87000","-450","1"],["36288000000000000","-10628640000000000","1275357600000000","-84095000000000","3416930000000","-90205500000","1577730000","-18150000","132000","-550","1"],["-3991680000000000000","1205438400000000000","-150917976000000000","10525807600000000","-459957300000000","13339535000000","-263755800000","3574230000","-32670000","192500","-660","1"],["479001600000000000000","-148644288000000000000","19315595520000000000","-1414014888000000000","65720683600000000","-2060701500000000","44990231000000","-692663400000","7494630000","-55770000","271700","-780","1"],["-62270208000000000000000","19802759040000000000000","-2659671705600000000000","203137530960000000000","-9957703756000000000","333611878600000000","-7909431530000000","135036473000000","-1666965300000","14744730000","-91091000","373100","-910","1"],["8717829120000000000000000","-2834656473600000000000000","392156797824000000000000","-31098926040000000000000","1597216056800000000000","-56663366760000000000","1440932292800000000","-26814537750000000","368411615000000","-3731227500000","27497470000","-143325000","500500","-1050","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-9","171","-4959","193401","-9476649","559122291","-38579438079","3047775608241","-271252029133449","26853950884211451","-2927080646379048159","348322596919106730921","-44933615002564768288809","6245772485356502792144451"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A048176","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["-10","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160480,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[5, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"5"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [5, +inf)","region_exact":null,"edges":["e-040c826eba4e7d74"]},{"id":160481,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-040c826eba4e7d74"]},{"id":160482,"position":2,"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-040c826eba4e7d74"]}],"region_exact":{},"incoming_edges":[{"id":"e-040c826eba4e7d74","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-1/5\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f45d29a8d57e6cc8","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":"8b05620c7a1e1566ec3f9f4f68cb703cfd7d714564f9ba122d8bde292ab33cfe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"5"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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-cdcdb3aca64a476d","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-4ef31ba37e5b2299","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."},{"id":"e-cdcdb3aca64a476d","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-4ef31ba37e5b2299","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."}],"used_by_total":2,"links":[{"id":"l-63c08b76093f3b91","kind":"oeis_match","node_a_id":"n-f45d29a8d57e6cc8","node_b_id":null,"fingerprint":null,"oeis_id":"A048176","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":38},"in_catalog":true,"oeis_id":"A048176","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A048176","relation":"identification"},{"oeis_id":"A048176","relation":"match_triangle"}],"bfiles":[{"node_id":"n-f45d29a8d57e6cc8","oeis_id":"A048176","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":38,"oeis_terms_agreeing":38,"oeis_terms_available":38,"term_count":990,"raw_size":41548}],"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:              -10,               1;\n  2:              200,             -30,               1;\n  3:            -6000,            1100,             -60,            1;\n  4:           240000,          -50000,            3500,         -100,            1;\n  5:        -12000000,         2740000,         -225000,         8500,         -150,         1;\n  6:        720000000,      -176400000,        16240000,      -735000,        17500,      -210,        1;\n  7:     -50400000000,     13068000000,     -1313200000,     67690000,     -1960000,     32200,     -280,     1;\n  8:    4032000000000,  -1095840000000,    118124000000,  -6728400000,    224490000,  -4536000,    54600,  -360,    1;\n  9: -362880000000000, 102657600000000, -11727000000000, 723680000000, -26932500000, 632730000, -9450000, 87000, -450, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10*n-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(10*n - x)*P[n - 1, 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] = -(10*n - 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A048176","target":"triangle","first_row":0,"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:              -10,               1;\n   3:              200,             -30,               1;\n   4:            -6000,            1100,             -60,            1;\n   5:           240000,          -50000,            3500,         -100,            1;\n   6:        -12000000,         2740000,         -225000,         8500,         -150,         1;\n   7:        720000000,      -176400000,        16240000,      -735000,        17500,      -210,        1;\n   8:     -50400000000,     13068000000,     -1313200000,     67690000,     -1960000,     32200,     -280,     1;\n   9:    4032000000000,  -1095840000000,    118124000000,  -6728400000,    224490000,  -4536000,    54600,  -360,    1;\n  10: -362880000000000, 102657600000000, -11727000000000, 723680000000, -26932500000, 632730000, -9450000, 87000, -450, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10*n-10-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[-(10*n - 10 - 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] = -(10*n - 10 - 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 (\u22121/5)x+1"],"edges":["e-040c826eba4e7d74"],"text":"A028338 \u2192 x \u21a6 (\u22121/5)x+1"},"headline":"A048176: Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","chain":"A028338 \u2192 x \u21a6 (\u22121/5)x+1"}