{"id":"n-31db8029f5c1d671","revision":0,"first_index":1,"fingerprint":"f6-acc791d1c44ba380","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= 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"],["3","2"],["11","14","4"],["45","84","48","8"],["197","484","416","144","16"],["903","2750","3160","1680","400","32"],["4279","15546","22380","16120","6000","1056","64"],["20793","87752","151872","138320","70560","19712","2688","128"],["103049","495368","1001728","1104320","718144","277760","60928","6656","256"],["518859","2798586","6475824","8385216","6640704","3298176","1010688","179712","16128","512"],["2646723","15827910","41246580","61368240","57355200","34882176","13816320","3459840","510720","38400","1024"],["13648869","89625756","259741680","436686360","470738400","339322368","165403392","53856000","11278080","1408000","90112","2048"],["71039373","508129452","1621094112","3039445200","3713838480","3098468736","1797210624","723469824","198063360","35312640","3784704","208896","4096"],["372693519","2884263798","10044902088","20781618000","28391570064","26926332576","18134907648","8718488064","2963380992","694160896","106895360","9957376","479232","8192"],["1968801519","16390465266","61875173724","140023292136","211548280944","224865697056","172616283840","96647041920","39364765440","11493081600","2336029696","314402816","25718784","1089536","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","185","1257","8925","65445","491825","3768209","29324405","231153133","1841801065","14810069497","120029657805","979470140661"],"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":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-5","-6"],["7","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-6","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":"e-d86ba9d5ebdfaca2","recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","8","8"],["-1","-4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-2"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176683,"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-9d1d77717d3adaeb"]},{"id":176684,"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-9d1d77717d3adaeb","e-a135b2bb55d9a2f7"]},{"id":176685,"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-9d1d77717d3adaeb","e-a135b2bb55d9a2f7"]},{"id":176686,"position":3,"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-9d1d77717d3adaeb","e-a135b2bb55d9a2f7"]},{"id":176687,"position":4,"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-a135b2bb55d9a2f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-a135b2bb55d9a2f7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-31db8029f5c1d671","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":"1f089e6fad99ad23eaeb6e2e26091a0a83e1d25efdf9ce5c8080d314734677b6","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-6f30da2298fd30ba","role":"sequence","name":"basis_map_A007318 of narayana","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-d86ba9d5ebdfaca2","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-31db8029f5c1d671","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":"5e9427fcd97be87f98ca982d2c176c3dd23058afcbdf27db5ab33c5aae50cd4b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-9d1d77717d3adaeb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-31db8029f5c1d671","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":"0a9cffc18e0e1f13bc1e4aa3b4a43444df100d1b39b249dabb7d81c783a988d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c3c79310b753c17e","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (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":[],"used_by_total":0,"links":[{"id":"l-14913afbc1df8ada","kind":"fingerprint_related","node_a_id":"n-31db8029f5c1d671","node_b_id":"n-36ea7ad71f537a76","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-36ea7ad71f537a76","other":{"id":"n-36ea7ad71f537a76","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["6","4"],["22","28","8"]],"card_rows":[["2"],["6","4"],["22","28","8"],["90","168","96","16"]],"row_sums_preview":["2","10","58","370","2514","17850","130890","983650"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["reverse","x \u21a6 x+1"],"edges":["e-a35c7072fcaf677b","e-9b41b0551a9c258e"],"text":"A114655 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A114655 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-188b545c88c24822","kind":"fingerprint_related","node_a_id":"n-40fe41bf121fa8b3","node_b_id":"n-31db8029f5c1d671","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-40fe41bf121fa8b3","other":{"id":"n-40fe41bf121fa8b3","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","14","11"]],"card_rows":[["1"],["2","3"],["4","14","11"],["8","48","84","45"]],"row_sums_preview":["1","5","29","185","1257","8925","65445","491825"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["x \u21a6 x/(x+1)"],"edges":["e-0286c892acd84d8a"],"text":"A114656 \u2192 x \u21a6 x/(x+1)"},"headline":"A114656 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3c04d82c173e61c6","kind":"fingerprint_related","node_a_id":"n-b6cda967a092feb1","node_b_id":"n-31db8029f5c1d671","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-b6cda967a092feb1","other":{"id":"n-b6cda967a092feb1","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["11","14","4"]],"card_rows":[["1"],["3","2"],["11","14","4"],["45","84","48","8"]],"row_sums_preview":["1","5","29","185","1257","8925","65445","491825"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8ce61d302cec295b","steps":["basis A007318"],"edges":["e-6804282af6df94ec"],"text":"A114687 \u2192 basis A007318"},"headline":"A114687 \u2192 basis A007318","chain":null}},{"id":"l-49c6118fd929b87b","kind":"fingerprint_related","node_a_id":"n-4a5cb5ea4703a9a6","node_b_id":"n-31db8029f5c1d671","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-4a5cb5ea4703a9a6","other":{"id":"n-4a5cb5ea4703a9a6","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","6"],["8","28","22"]],"card_rows":[["2"],["4","6"],["8","28","22"],["16","96","168","90"]],"row_sums_preview":["2","10","58","370","2514","17850","130890","983650"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["x \u21a6 x/(x+1)"],"edges":["e-460d39e35063c243"],"text":"A114655 \u2192 x \u21a6 x/(x+1)"},"headline":"A114655 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7a1d9ae1a3349c75","kind":"fingerprint_related","node_a_id":"n-31db8029f5c1d671","node_b_id":"n-31a1b9b1779ea532","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-31a1b9b1779ea532","other":{"id":"n-31a1b9b1779ea532","name":"basis_map_A013609 of shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","first_index":0,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["33","42","12"]],"card_rows":[["1"],["6","4"],["33","42","12"],["180","336","192","32"]],"row_sums_preview":["1","10","87","740","6285","53550","458115","3934600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-984320118b653f6b","steps":["basis A007318","basis A013609"],"edges":["e-2345ffa617ccabd7","e-58c7c138b5508493"],"text":"A132812 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A132812 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-7cbd5561632584fc","kind":"fingerprint_related","node_a_id":"n-31db8029f5c1d671","node_b_id":"n-564c31896f63b482","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-564c31896f63b482","other":{"id":"n-564c31896f63b482","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","14","11"]],"card_rows":[["1"],["2","3"],["4","14","11"],["8","48","84","45"]],"row_sums_preview":["1","5","29","185","1257","8925","65445","491825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8ce61d302cec295b","steps":["basis A007318","reverse"],"edges":["e-6804282af6df94ec","e-302c57842721c235"],"text":"A114687 \u2192 basis A007318 \u2192 reverse"},"headline":"A114687 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-9c06d288bcf3fa59","kind":"fingerprint_related","node_a_id":"n-bd61f7a659e54e52","node_b_id":"n-31db8029f5c1d671","fingerprint":"f6-acc791d1c44ba380","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acc791d1c44ba380"},"other_id":"n-bd61f7a659e54e52","other":{"id":"n-bd61f7a659e54e52","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-acc791d1c44ba380","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","14","11"]],"card_rows":[["1"],["2","3"],["4","14","11"],["8","48","84","45"]],"row_sums_preview":["1","5","29","185","1257","8925","65445","491825"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8ce61d302cec295b","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-0ad5e2ffb4ffd4ba","e-34ee51153b1773b2"],"text":"A114687 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A114687 \u2192 reverse \u2192 x \u21a6 x/(x+1)","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:      3,       2;\n   3:     11,      14,       4;\n   4:     45,      84,      48,       8;\n   5:    197,     484,     416,     144,      16;\n   6:    903,    2750,    3160,    1680,     400,      32;\n   7:   4279,   15546,   22380,   16120,    6000,    1056,      64;\n   8:  20793,   87752,  151872,  138320,   70560,   19712,    2688,    128;\n   9: 103049,  495368, 1001728, 1104320,  718144,  277760,   60928,   6656,   256;\n  10: 518859, 2798586, 6475824, 8385216, 6640704, 3298176, 1010688, 179712, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (7*n-5+(6*n-6)*x)*P(n-1,x) - (2+6*x+4*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[((7*n - 5 + (6*n - 6)*x)*P[n - 1, x] - (2 + 6*x + 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];\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] = ((7*n - 5 + (6*n - 6)*x)*v[i - 1] - (2 + 6*x + 4*x^2)*deriv(v[i - 1], x))/(n + 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}}],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["reverse","x \u21a6 x+1"],"edges":["e-87267df0bda73e27","e-9d1d77717d3adaeb"],"text":"A114656 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A114656 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}
